Totally Splittable Polytopes

TOT ALL Y SPLITT ABLE POL YTOPES SVEN HERRMANN AND MICHAEL JOSWIG Abstra t. A split of a p olytop e is a (neessarily regular) sub division with exatly t w o maximal ells. A p olytop e is total ly splittable if ea h triangulation (without additional v erties) is a ommon renemen t of splits. This pap er establishes a omplete lassiation of the totally splittable p olytop es. 1. Intr odution Splits (of h yp ersimplies) rst o urred in the w ork of Bandelt and Dress on deomp o- sitions of nite metri spaes with appliations to ph ylogenetis in algorithmi biology [ 1℄. This w as later generalized to a result on arbitrary p olytop es b y Hirai [ 10 ℄ and the au- thors [9℄. While man y p olytop es do not admit a single split, the purp ose of this pap er is to study p olytop es with v ery man y splits. The set of all regular sub divisions of a p olytop e P , partially ordered b y renemen t, has the struture of the fae lattie of a p olytop e, the se  ondary p olytop e of P in tro dued b y Gel ′ fand, Kaprano v, and Zelevinsky [6℄. The v erties of the seondary p olytop e or- resp ond to the regular triangulations, while the faets orresp ond to the regular oarsest sub divisions. There is a host of kno wledge on triangulations of p olytop es [5℄, but infor- mation on oarsest sub divisions is sare. Splits are ob viously oarsest sub divisions and moreo v er kno wn to b e regular. So they orresp ond to faets of the seondary p olytop e. The total splittabilit y of P is equiv alen t to the prop ert y that ea h faet of the seondary p olytop e of P arises from a split. Via a  omp atibility relation the splits of a p olytop e form an abstrat simpliial omplex. F or instane, for the h yp ersimplies ∆( d, n ) this turns out to b e a sub omplex of the Dr essian Dr ( d, n ) whi h is an outer appro ximation (in terms of matroid deomp ositions) of the tropial v ariet y arising from the Grassmannian of d -planes in n -spae; see [9, Theorem 7.8℄ and [8℄. As an b e exp eted the assumption of total splittabilit y restrits the om binatoris of P drastially . W e pro v e that the totally splittable p olytop es are the simplies, the p olygons, the regular rossp olytop es, the prisms o v er simplies, or joins of these. In ter- estingly , our lassiation seems to oinide with those innite families of p olytop es for whi h the seondary p olytop es are kno wn. This suggests that, in order to deriv e more detailed information ab out the seondary p olytop es of other p olytop es, it is ruial to systematially in v estigate oarsest sub divisions other than splits. Su h a task, ho w ev er, is b ey ond the sop e of this pap er. This is ho w our pro of (and th us the pap er) is organized: It will frequen tly turn out to b e on v enien t to phrase fats in terms of a Gale dual of a p olytop e. Hene w e b egin our pap er with a short in tro dution to Gale dualit y and  ham b er omplexes. The rst imp ortan t step to w ards the lassiation is the easy Prop osition 12 whi h sho ws that the neigh b ors of a v ertex of a totally splittable p olytop e m ust span an ane h yp erplane. Then the follo wing observ ation turns out to b e useful: Whenev er P is a prism o v er a Date : No v em b er 9, 2018. Sv en Herrmann is supp orted b y a Graduate Gran t of TU Darmstadt. Resear h b y Mi hael Joswig is supp orted b y DF G Resear h Unit P olyhedral Surfaes. 1 2 HERRMANN AND JOSWIG ( d − 1) -simplex or a d -dimensional regular rossp olytop e with d ≥ 3 , there is no plae for a p oin t v outside P su h that con v ( P ∪ { v } ) is totally splittable. In this sense, prisms and rossp olytop es are maximal ly total ly splittable . It is lear that the ase of d = 2 is quite dieren t; and it is one te hnial diult y in the pro of to in trinsially distinguish b et w een p olygons and higher dimensional p olytop es. The next step is a areful analysis of the Gale dual of a totally splittable p olytop e whi h mak es it p ossible to reognize a p oten tial deomp osition as a join. A nal redution argumen t allo ws one to onen trate on maximally totally splittable fators, whi h then an b e iden tied again via their Gale duals. W e are indebted to the anon ymous referees for v ery areful reading whi h lead to sev eral impro v emen ts in the exp osition. 2. Splits and Gale Duality Let V b e a onguration of n ≥ d + 1 (not neessarily distint) non-zero v etors in R d +1 whi h linearly spans the whole spae. Often w e iden tify V with the n × ( d + 1) -matrix whose ro ws are the p oin ts in V , and our assumption sa ys that the matrix V has full rank d + 1 . Su h a v etor onguration giv es rise to an oriente d matr oid in the follo wing w a y: F or a linear form a ∈ ( R d +1 ) ⋆ w e ha v e a  ove tor C ⋆ ∈ { 0 , + , −} V b y C ⋆ ( v ) :=      0 if av = 0 , + if av > 0 , − if av < 0 . F or ǫ ∈ { 0 , + , −} w e let C ⋆ ǫ := { v ∈ V | C ⋆ ( v ) = ǫ } , and w e all the m ultiset C ⋆ + ∪ C ⋆ − the supp ort of C ⋆ . Oasionally , the omplemen t C ⋆ 0 will b e alled the  osupp ort of C ⋆ . A o v etor whose supp ort is minimal with resp et to inlusion of m ultisets is a  o ir uit ; equiv alen tly , its osupp ort is maximal. Dually , C ⊂ { 0 , + , −} V is alled a ve tor of V if the linear dep endene X v ∈ C + λ v v = X v ∈ C − λ v v holds for some o eien ts λ v > 0 ; here C ǫ is dened as for the o iruits. The v etors with minimal supp ort are the ir uits . Note that a p oin t onguration denes the iruits and o iruits only up to a sign rev ersal. Oasionally , w e will sp eak of unique (o-)iruits with giv en prop erties, and in these ases w e alw a ys mean uniqueness up to su h a rev ersal of the signs. See monograph [4℄ for all details and pro ofs of prop erties of orien ted matroids. No w onsider an n × ( n − d − 1) -matrix V ⋆ of full rank n − d − 1 satisfying V T V ⋆ = 0 ; that is, the olumns of V ⋆ form a basis of the k ernel of V T . Then the onguration of ro w v etors of V ⋆ is alled a Gale dual of V . An y Gale dual of V is uniquely determined up to ane equiv alene. Ea h v etor v ∈ V orresp onds to a ro w v etor v ⋆ of V ⋆ , alled the ve tor dual to v . Throughout w e will assume that all dual v etors are either zero or ha v e unit Eulidean length. If v ⋆ is zero then all v etors other than v span a linear h yp erplane not on taining v . W e all V pr op er if V ⋆ do es not on tain an y zero v etors. In the primal view, this means that con v V is not a p yramid. F or the remainder of this setion w e will assume that V is prop er whene V ⋆ an b e iden tied with a onguration of n p oin ts on the unit sphere S n − d − 2 . Notie that these n p oin ts are not neessarily pairwise distint. Rep etitions ma y o ur ev en if the v etors in V are pairwise distint. The onnetion b et w een Gale dualit y and orien ted matroids is the follo wing: The ir- uits of V are preisely the o iruits of V ⋆ and on v ersely . W e dene the oriente d matr oid of V as its set of o iruits. Anely equiv alen t v etor ongurations ha v e the same orien ted matroid, but the on v erse do es not hold. TOT ALL Y SPLITT ABLE POL YTOPES 3 No w let P b e a d -dimensional p olytop e in R d with n v erties. By homogenizing the v erties V ert P , w e obtain a onguration V P of n non-zero v etors in R d +1 whi h linearly spans the whole spae. The o iruits of V P are giv en b y the linear h yp erplanes spanned b y v etors in V P . The v etor onguration V P is prop er if and only if P is not a p yramid, and w e will assume that this is the ase. The Gale dual of P is the spherial p oin t onguration Gale( P ) := V ⋆ P , whi h again is unique up to (spherial) ane equiv alene. 1 2 3 4 5 1 2 3 4 5 Figure 1. P en tagon and Gale dual. Corresp onding v erties and dual v e- tors are lab eled alik e. Example 1. The matries V :=       1 1 0 1 0 2 1 − 1 1 1 − 1 0 1 0 − 1       and V ⋆ :=       − 1 / 3 − 1 2 / 3 1 − 4 / 3 − 1 1 0 0 1       are Gale duals of ea h other. The ro ws of the matrix V are the homogenized v erties of the p en tagon sho wn to the left in Figure 1. The Gale dual obtained from pro jeting V ⋆ to S 1 is sho wn to the righ t. W e are in terested in p olytopal sub divisions of our p olytop e P and in tend to study them via Gale dualit y . This requires the in tro dution of some notation. A p olytopal sub division of P is r e gular if it is indued b y a lifting funtion on the v erties of P . The set of all lifting funtions λ ∈ R n induing a xed regular sub division Σ λ is a relativ ely op en p olyhedral one in R n , the se  ondary  one of Σ λ . The set of all seondary ones forms a p olyhedral fan, the se  ondary fan SecF an( P ) . It turns out that the seondary fan is the normal fan of a p olytop e of dimension n − d − 1 , and an y su h p olytop e is a se  ondary p olytop e of P , that is the seondary p olytop e SecP o ly( P ) is dened only up to normal equiv alene. The v erties of SecP o ly( P ) orresp ond to the regular triangulations of P . The redution in dimension omes from the fat that all the seondary ones in SecF an( P ) ha v e a ( d + 1) -dimensional linealit y spae in ommon. By fatoring out this linealit y spae and in terseting with the unit sphere one obtains the spherial p olytopal omplex SecF an ′ ( P ) in S n − d − 2 . It is dual to the b oundary omplex of the seondary p olytop e. 4 HERRMANN AND JOSWIG No w x a Gale dual G := Gale( P ) . Ea h subset I ⊆ [ n ] orresp onds to a set of (homogenized) v erties V I . W e set I ⋆ := [ n ] \ I and V ⋆ I := { v ⋆ i | i ∈ I } . Then the set V I anely spans R d if and only if the duals of the omplemen t, that is, the set V ⋆ I ⋆ = { v ⋆ i | i ∈ [ n ] \ I } is linearly indep enden t. In partiular, for ea h d -dimensional simplex con v V J with # J = d + 1 the set p os V ⋆ J ⋆ ∩ S n − d − 2 is a full-dimensional spherial simplex, whi h is alled the dual simplex of con v V J . The hamb er  omplex Cham b er( P ) is the set of subsets of S n − d − 2 arising from the in tersetions of all the dual simplies. The follo wing theorem b y Billera, Gel ′ fand, and Sturmfels [3℄ (see also [5, 5.3℄) is essen tial. Theorem 2 ([3 , Theorem 3.1℄) . The hamb er  omplex Cham b er( P ) is anti-isomorphi to the b oundary  omplex of the se  ondary p olytop e SecP o ly( P ) . A split of the p olytop e P is a p olytopal deomp osition (without new v erties) with exatly t w o maximal ells. Splits are alw a ys regular. The ane h yp erplanes w eakly sep- arating the t w o maximal ells of a split are  haraterized b y the prop ert y that they do not ut through an y edges of P [ 9, Observ ation 3.1℄; they are alled split hyp erplanes . T w o splits of P are  omp atible if their split h yp erplanes do not in terset in the in terior of P . They are we akly  omp atible if they admit a ommon renemen t. Clearly , om- patibilit y implies w eak ompatibilit y , but the on v erse is not true; see Example 3 b elo w. By denition the splits are oarsest sub divisions of P and hene orresp ond to ra ys in the seondary fan or, equiv alen tly , to faets of the seondary p olytop e and to v erties in the  ham b er omplex. The split  omplex Split( P ) is the abstrat ag-simpliial omplex whose v erties are the splits of P whi h is indued b y the ompatibilit y relation. The we ak split  omplex Split w ( P ) is the sub omplex of SecF an ′ ( P ) indued b y the splits. Example 3. Let P = con v {± e i | i ∈ [ d ] } b e a regular rossp olytop e in dimension d . The splits of P are giv en b y the o ordinate h yp erplanes x i = 0 , for i ∈ [ d ] . By om bining an y d − 1 of these splits one gets a triangulation of P . This sho ws that the w eak split omplex is isomorphi to the b oundary of a ( d − 1) -simplex. Ho w ev er, an y t w o o ordinate h yp erplanes on tain the origin, whene the orresp onding splits are not ompatible. The split omplex of P has d isolated p oin ts. See also [9, Example 4.9℄. Prop osition 4. The split  omplex Split( P ) and the we ak split  omplex Split w ( P ) of a p olytop e P only dep end on the oriente d matr oid of P . Pr o of. Ea h split S of P denes a o iruit C ⋆ of the orien ted matroid of P . A h yp erplane whi h separates P denes a split if and only if it do es not separate an y edge of P . Ho w ev er, an edge of P is a o v etor of P with exatly t w o p ositiv e en tries and it is separated b y S if and only if one if the en tries is in C ⋆ + and the other is in C ⋆ − . So one sees that the set of splits of P only dep ends on the orien ted matroid of P . No w it remains to sho w that also the ompatibilit y and w eak ompatibilit y relations among splits only dep end on the orien ted matroid. Let S 1 and S 2 b e t w o splits of P with split h yp erplanes H S 1 and H S 2 , resp etiv ely . Supp ose that S 1 and S 2 are inompatible. Then there exists a p oin t x ∈ in t P ∩ H S 1 ∩ H S 2 . Sine b oth split h yp erplanes are spanned b y v erties of P and sine, moreo v er, ea h split h yp erplane do es not in terset an y edge the p oin t x is a on v ex om bination of v erties of P on H S 1 as w ell as a on v ex om bination of v erties of P on H S 2 . Th us x giv es rise to a v etor C in the orien ted matroid of P su h that C + is supp orted on v erties of P lying on H S 1 and C − is supp orted on v erties of P lying on H S 2 . That x is on tained in the in terior of P is equiv alen t to the prop ert y that C + ∪ C − is not on tained in an y faet of TOT ALL Y SPLITT ABLE POL YTOPES 5 P . Sine the faets are preisely the p ositiv e o iruits it follo ws that this an b e read o from the orien ted matroid of P . The statemen t for the w eak split omplex follo ws from the fat that one an onstrut ommon renemen ts of giv en sub divisions while only kno wing the orien ted matroid of the underlying p olytop e [5, Corollary 4.1.43℄.  Note that, of ourse, kno wing the om binatoris, that is the fae lattie of a p olytop e is not enough for kno wing its split omplex or ev en its splits. As an example onsider the regular o tahedron (with three splits; see Example 3) and an o tahedron with p er- turb ed v erties (whi h do es not ha v e an y split). F urther, note that the set of regular sub divisions of a p olytop e do es not only dep end on the orien ted matroid but rather on the o ordinatization. So the split sub divisions form a subset of all regular sub divisions whi h is indep enden t of the o ordinatization. In partiular, the split omplex is a ommon appro ximation for the seondary fans of all p olytop es with the same orien ted matroid but anely inequiv alen t o ordinates. The next lemma explains ho w splits an b e reognized in the  ham b er omplex. W e on tin ue to use the notation in tro dued ab o v e. In partiular, P is the p olytop e and G its spherial Gale dual. Lemma 5. A p oint x ∈ S n − d − 2 denes a split of P if and only if ther e exists a unique ir uit C in G suh that p os x = p os V ⋆ C + ∩ p os V ⋆ C − . Pr o of. Consider x ∈ S n − d − 2 su h that its  ham b er is dual to a split S of P . Then the split h yp erplane H S denes a unique o iruit C of P . Equiv alen tly , C is a iruit of G . Moreo v er, p os V ⋆ C + and p os V ⋆ C − orresp ond to the t w o maximal ells of S , and p os x = p os V ⋆ C + ∩ p os V ⋆ C − . Supp ose that there is another iruit C ′ in G with the same prop ert y . Then the h yp erplane H dened b y the elemen ts of the orresp onding o iruit of V P separates the preimage of x from all remaining v erties of P . Ho w ev er, sine x denes a split S w e get H = H S and hene the uniqueness. Con v ersely , let C b e the unique iruit of G su h that p os x = p os V ⋆ C + ∩ p os V ⋆ C − for some x ∈ S n − d − 2 . Ob viously , x is a ra y of the  ham b er omplex, and hene it is dual to a oarsest sub division S of P . By [3, Lemma 3.2℄, the sub division orresp onding to x has t w o maximal ells, sine p os V ⋆ C + and p os V ⋆ C − are the only (neessarily minimal) dual ells on taining x .  Example 6. Let P b e the p en tagon and G its Gale dual from Example 1. Then C = (0 + 0 − − ) is a o iruit of P orresp onding to the split dened b y the line through the v erties v 1 and v 3 . Clearly , C is also a iruit of G , with C + = { 2 } and C − = { 4 , 5 } . W e ha v e p os v ⋆ 2 = p os V ⋆ { 2 } ∩ p os V ⋆ { 4 , 5 } , and C is the unique iruit of G yielding p os v ⋆ 2 as the in tersetion of its p ositiv e and its negativ e one. The t w o maximal ells of the split are the quadrangle con v V { 2 } ⋆ and the triangle con v V { 4 , 5 } ⋆ . See Figure 1. With ea h split S of P w e asso iate the unique iruit C [ S ] of G from Lemma 5. If V ⋆ C [ S ] + or ( V ⋆ C [ S ] − ) onsists of a single elemen t v ⋆ orresp onding to a v ertex v of P , w e all S the vertex split for the v ertex v and also write C [ v ] for C [ S ] . Note that the supp ort of C [ v ] orresp onds to the set of all v erties of P that are onneted to v b y an edge. Lemma 7. L et S and S ′ b e vertex splits with r esp e t to verti es v and v ′ of P . Then S and S ′ ar e  omp atible if and only if v and v ′ ar e not joine d by an e dge. Pr o of. It is easily seen that t w o splits S, S ′ are ompatible if and only if (p ossibly after the negation of one or b oth of the iruits) C [ S ] + ⊆ C [ S ′ ] + and C [ S ′ ] − ⊆ C [ S ] − . F or a v ertex split with resp et to the v ertex v w e ha v e C [ v ] + = { v ⋆ } or C [ v ] − = { v ⋆ } . Ho w ev er, if v and v ′ are joined b y an edge, then v ⋆ ∈ C [ v ′ ] 0 , so the ab o v e onditions annot hold. 6 HERRMANN AND JOSWIG On the other hand, if v and v ′ are not joined b y an edge, and, sa y , C [ v ] + = { v ⋆ } , then (p ossibly after a negation) v ⋆ ∈ C [ v ′ ] + whi h implies { v ⋆ } = C [ v ] + ⊆ C [ v ′ ] + .  Clearly , P admits a v ertex split at the v ertex v if and only if the neigh b ors of v in the v ertex-edge graph of P lie on a ommon h yp erplane. In partiular, if P is simple then ea h v ertex giv es rise to a v ertex split. 3. Tot all y Splitt able Pol ytopes W e all a p olytop e total ly splittable if all regular triangulations of P are split triangu- lations. W e aim at the follo wing omplete  haraterization. Theorem 8. A p olytop e P is total ly splittable if and only if it has the same oriente d matr oid as a simplex, a r ossp olytop e, a p olygon, a prism over a simplex, or a (p ossibly multiple) join of these p olytop es. By Prop osition 4 the set of splits and their (w eak) ompatibilit y only dep ends on the orien ted matroid of P , and hene the notion totally splittable also dep ends on the orien ted matroid only . The join P ∗ Q of a d -p olytop e P and an e -p olytop e Q is the on v ex h ull of P ∪ Q , seen as subp olytop es in m utually sk ew ane subspaes of R d + e +1 . F or instane, a 3 -simplex is the join of an y pair of its disjoin t edges. In order to a v oid um b ersome notation in the remainder of this setion w e do not distinguish b et w een an y t w o p olytop es sharing the same orien ted matroid. F or instane,  P is a join of P 1 and P 2  atually means  P has the same orien ted matroid as the join of P 1 and P 2  and so on. Example 9. W e insp et the lasses of p olytop es o urring in Theorem 8. (i) Simplies are totally splittable in a trivial w a y . (ii) A triangulation of an n -gon is equiv alen t to  ho osing n − 3 diagonals whi h are pairwise non-in terseting. This is a ompatible system of splits, and hene ea h p olygon is totally splittable; see [9, Example 4.8℄. The seondary p olytop e of an n -gon is the ( n − 3) -dimensional asso iahedron [6 , Chapter 7, 3.B℄. (iii) Let P = conv {± e i | i ∈ [ d ] } b e a regular rossp olytop e in dimension d as in Example 3 . The splits orresp ond to the o ordinate h yp erplanes, and an y d − 1 of them indue a triangulation of P . Con v ersely , ea h triangulation of P arises in this w a y . See [9 , Example 4.9℄. A Gale dual of P is giv en b y the m ultiset G ⊂ S d − 2 onsisting of all p oin ts  e i   i ∈ [ d − 1]  ∪  − 1 √ d − 1 d − 1 X i =1 e i  , where ea h p oin t o urs exatly t wie. All the v erties in the  ham b er omplex orresp ond to v ertex splits, and the  ham b er omplex is the normal fan of a ( d − 1) -simplex (where ea h v ertex arries t w o lab els). So the seondary p olytop e of P is a ( d − 1) -simplex. See Figure 2 (left) b elo w for d = 3 . (iv) Let P b e the prism o v er a ( d − 1) -simplex. Then the dual graph of an y tri- angulation of P is a path with d no des. The seondary p olytop e of P is the ( d − 1) -dimensional p erm utohedron [6 , Chapter 7, 3.C℄. See Figure 2 (righ t) b elo w for d = 3 . Remark 10. As the seondary p olytop e of a join of p olytop es is the pro dut of their seondary p olytop es (e.g., this an b e inferred from [ 5 , Corollary 4.2.8℄), Theorem 8 and Example 9 sho w that the seondary p olytop es of totally splittable p olytop es are (p ossibly m ultiple) pro duts of simplies, p erm utohedra, and asso iahedra. TOT ALL Y SPLITT ABLE POL YTOPES 7 1 2 3 4 5 6 1 2 3 4 5 6 Figure 2. Gale diagrams of the regular o tahedron (left) and of the prism o v er a triangle (righ t). Remark 11. One an ask the question: What is the t ypial b eha vior of a p olytop e in terms of splits? The smallest example of a p olytop e that do es not ha v e an y split is giv en b y an o tahedron whose v erties are sligh tly p erturb ed in to general p osition. Moreo v er, an y 2 -neighb orly p olytop e (that is, an y t w o v erties share an edge) do es not admit an y split [9, Prop osition 3.4℄. On the other hand, d -dimensional simple p olytop es with n v erties ha v e at least n splits: Ea h v ertex is onneted to exatly d other v erties whi h span a split h yp erplane for the orresp onding v ertex split. This sho ws that the answ er of the seemingly more preise question of ho w man y splits is a random p olytop e exp eted to ha v e highly dep ends on the  hosen mo del. On the one hand, a d -p olytop e whose faets are  hosen uniformly at random tangen t to the unit sphere is simple with probabilit y one; hene it has at least as man y splits as v erties. On the other hand one an  ho ose mo dels su h that the p olytop es generated are 2 -neigh b orly with high probabilit y [ 11℄; su h p olytop es do not ha v e an y splits. It is ob vious that total splittabilit y is a sev ere restrition among p olytop es. The follo w- ing result is a k ey rst step. As an essen tial to ol w e use that an y ordering of the v erties of a p olytop e indues a triangulation, the plaing triangulation with resp et to that or- dering [5, 4.3.1℄. Moreo v er, suessiv e plaing of new v erties an b e used to extend an y triangulation of a subp olytop e. Prop osition 12. L et P b e a total ly splittable p olytop e. Then e ah fa e, e ah vertex gur e, and e ah subp olytop e Q := con v ( V \ { v } ) for a vertex v ∈ V is total ly splittable. Mor e over, v gives rise to a vertex split, and the neighb ors of v sp an a fa et of Q . Pr o of. Let ∆ b e an arbitrary triangulation of a faet F of P . W e ha v e to sho w that ∆ is indued b y splits of F . By plaing the v erties of P not in F in an arbitrary order w e an extend ∆ to a triangulation ∆ ′ of P . As P is totally splittable ∆ ′ is indued b y splits of P . A split of P either do es not separate F , or it is a split of F . This implies that ∆ is indued b y splits of F . Indutiv ely , this sho ws the total splittabilit y of all faes of P . Consider the subp olytop e Q := conv( V \ { v } ) for some v ertex v of P . W e an assume that P is not a simplex, whene Q is full-dimensional. T ak e an arbitrary triangulation Σ of Q . By plaing v this extends to a triangulation Σ ′ of P . The d -simplies of Σ ′ on taining v are the ones (with ap ex v ) o v er those o dimension ( d − 1) -faes of Σ whi h span a h yp erplane w eakly separating Q from v . By assumption, Σ ′ is a split triangulation, and hene ea h in terior ell of o dimension one spans a split h yp erplane. Fix a d -simplex σ ∈ Σ ′ on taining v . The faet of σ not on taining v is an in terior ell of o dimension 8 HERRMANN AND JOSWIG one, whi h is wh y it spans a split h yp erplane H . Sine H annot ut through the other simplies in Σ ′ all neigh b ors of v in the v ertex-edge graph of P are on tained in H . This pro v es that H is the split h yp erplane of the v ertex split to v , and H in tersets Q in a faet. This also sho ws that the triangulation Σ of Q is indued b y splits of Q , and Q is totally splittable. The v ertex gure of P at v is anely equiv alen t to the faet Q ∩ H of Q , and hene the total splittabilit y of the v ertex gure follo ws from the ab o v e.  Remark 13. The same argumen t as in the pro of ab o v e sho ws: Ea h h yp erplane spanned b y d anely indep enden t v erties of a totally splittable p olytop e denes a faet or a split. Note that there exist p olytop es for whi h ea h v ertex denes a v ertex split, but whi h are not totally splittable. An example is the 3 -ub e whi h is simple, and hene ea h v ertex denes a v ertex split [ 9, Remark 3.3℄, but whi h has sev eral triangulations whi h are not indued b y splits [ 9, Examples 3.8 and 4.10℄. It is ruial that, b y Prop osition 12, the neigh b ors of a v ertex v of a totally splittable p olytop e span a h yp erplane, whi h w e denote b y v ⊥ . t w o v erties of P are neighb ors if they share an edge w in the v ertex-edge graph of P . Prop osition 12 mak es it p ossible to re-read Lemma 5 as follo ws. Corollary 14. L et v b e a vertex of a total ly splittable p olytop e P . Then v ∈ \ w neighb or vertex to v w ⊥ . Remark 15. In the situation of Prop osition 12 all faets of Q are also faets of P exept for the faet F spanning the h yp erplane v ⊥ . Moreo v er, all v erties of Q are also v erties of P . In this situation w e sa y that v is almost b eyond the faet F of Q . This is sligh tly more general than requiring v to b e b eyond Q , whi h means that F is the unique faet of Q violated b y v , and additionally v is not on tained in an y h yp erplane spanned b y a faet of Q . That F is violate d b y v means that the losed ane halfspae with b oundary h yp erplane aff F do es not on tain the p oin t v . If v is b ey ond F and d = dim P = dim Q ≥ 3 then the v ertex-edge graph of Q is the subgraph of the v ertex-edge graph of P indued on V ert P \ { v } = V ert Q . The neigh b ors of v are preisely the v erties on the faet F of Q . Lemma 16. F or two p olytop es P and Q the join P ∗ Q is total ly splittable if and only if b oth P and Q ar e. Pr o of. Supp ose that P ∗ Q is totally splittable. Then P and Q b oth o ur as faes of P ∗ Q , and the laim follo ws from Prop osition 12 . Let dim P = d and dim Q = e , and assume that P and Q b oth are totally splittable. The join of a d -simplex and an e -simplex is a ( d + e + 1) -simplex, and hene the join ell-b y-ell of a triangulation of P and a triangulation of Q yields a triangulation of P ∗ Q . Con v ersely , ea h triangulation of P ∗ Q arises in this w a y [ 5, Theorem 4.2.7℄. The join of a split h yp erplane of P with aff Q and the join of a split h yp erplane of Q with aff P yields split h yp erplanes of P ∗ Q . No w onsider an y triangulation ∆ of P ∗ Q . Then there are triangulations ∆ P and ∆ Q of P and Q , resp etiv ely , su h that ∆ = ∆ P ∗ ∆ Q . By assumption, there is a set S P of splits of P induing ∆ P . Lik ewise S Q is the set of splits induing ∆ Q . Then the set of joins of all splits from S P with aff Q (as an ane subspae of R d + e +1 ) and the set of joins of all splits from S Q with aff P join tly indue the triangulation ∆ .  Lemma 16 together with Example 9 ompletes the pro of that all the p olytop es listed in Theorem 8 are, in fat, totally splittable. The remainder of this setion is dev oted to pro ving that there are no others. TOT ALL Y SPLITT ABLE POL YTOPES 9 Prop osition 17. L et P ⊂ R d b e a pr op er total ly splittable d -p olytop e. Then P is a r e gular r ossp olytop e if and only if the interse tion T v ∈ V ert P v ⊥ is not empty. Pr o of. Clearly , the regular rossp olytop e P = con v {± e i | i ∈ [ d ] } has the prop ert y that the in tersetion of its split h yp erplanes is the origin. Con v ersely , supp ose that P is not a rossp olytop e. W e assumed that P is prop er, meaning that P is not a p yramid. Hene there exists a v ertex v of P su h that at least t w o v erties u, w are separated from v b y the h yp erplane v ⊥ . By Prop osition 12, the split h yp erplane v ⊥ passes through the neigh b ors of v in the v ertex-edge graph of P . Sine u is on the same side of v ⊥ as w it follo ws that v ⊥ 6 = w ⊥ and, moreo v er, v ⊥ ∩ w ⊥ ∩ in t P = ∅ . No w supp ose that the in tersetion of all split h yp erplanes on tains p oin ts in the b oundary of P . But sine the split h yp erplanes do not ut through edges, the in tersetion m ust on tain at least one v ertex x ∈ V ert P . This is a on tradition sine x 6∈ x ⊥ . By a similar argumen t, w e an exlude the nal p ossibilit y that the in tersetion of all split h yp erplanes on tains an y p oin ts outside P . Therefore this in tersetion is empt y , as w e w an ted to sho w.  In a w a y rossp olytop es (whi h are not quadrangles) are maximally totally splittable. Lemma 18. L et P ⊂ R d b e a d -dimensional r e gular r ossp olytop e and v ∈ R d \ P b e a p oint almost b eyond the fa et F of P . If d ≥ 3 then con v( P ∪ { v } ) is not total ly splittable. Pr o of. Without loss of generalit y P = conv {± e 1 , ± e 2 , . . . , ± e d } . Supp ose that con v( P ∪ { v } ) is totally splittable. Sine w e assumed d ≥ 3 ea h v ertex w of P has at least d + 1 neigh b ors. A t least d anely indep enden t v erties among these are still neigh b ors of w in con v ( P ∪ { v } ) , so the h yp erplane w ⊥ with resp et to P is the same as w ⊥ with resp et to con v ( P ∪ { v } ) . W e ha v e that F ⊥ := T w ∈ V ert F w ⊥ = { 0 } , whi h implies v 6∈ F ⊥ , a on tradition to Corollary 14.  Figure 3. Con v ex h ull of prism plus one p oin t almost b ey ond a quadran- gular faet, v ertex-edge graph (left) and a non-split triangulation (righ t). The same onlusion as in Lemma 18 holds for prisms o v er simplies as w ell. See also Figure 3 and Example 20 b elo w. Lemma 19. L et P ⊂ R d b e a prism over a ( d − 1) -simplex and v ∈ R d \ P a p oint whih is almost b eyond a fa et F of P . If d ≥ 3 then con v( P ∪ { v } ) is not total ly splittable. Pr o of. Supp ose that con v ( P ∪ { v } ) is totally splittable. As in the pro of of Lemma 18 w e are aiming at a on tradition to Corollary 14. First supp ose that v is b ey ond F , and hene for w ∈ V ert P the h yp erplanes w ⊥ with resp et to P and con v( P ∪ { v } ) 10 HERRMANN AND JOSWIG oinide, sine d ≥ 3 ; see Remark 15. Up to an ane transformation w e an assume that P = con v { e 1 , e 2 , . . . , e d , f 1 , f 2 , . . . , f d } with f k = − X i 6 = k e i . The neigh b ors of the v ertex e k are e 1 , e 2 , . . . , e k − 1 , e k +1 , . . . , e d and f k ; symmetrially for the f k . A diret omputation sho ws that e ⊥ k = { x | x k = 0 } and f ⊥ k = ( x      2 X i 6 = k x i = ( d − 2)( x k − 1) ) . W e ha v e to distinguish t w o ases: the faet F of P violated b y v ma y b e a ( d − 1) -simplex or a prism o v er a ( d − 2) -simplex. If F is a simplex, for instane, con v { e 1 , e 2 , . . . , e d } , then w e an onlude that the set T w ∈ F w ⊥ = { 0 } whi h is in the in terior of P and hene annot b e equal to v . If, ho w ev er, F is a prism, for instane, with the v erties e 1 , e 2 , . . . , e d − 1 , f 1 , f 2 , . . . , f d − 1 , w e an ompute that \ w ∈ V ert F w ⊥ =  2 − d 2 e d  , again an in terior p oin t. In b oth ases w e arriv e at the desired on tradition to Corol- lary 14. No w supp ose that v violates F but it is not b ey ond F , that is, v is on tained in the ane h ull of some faet F ′ of P . Let us assume that d ≥ 4 and that the assertion is true for d = 3 . Then the p olytop e con v( F ′ ∪ { v } ) is totally splittable b y Prop osition 12. Again, F ′ ma y b e a ( d − 1) -simplex or a prism o v er a ( d − 2) -simplex. If F ′ is a ( d − 1) -simplex, it an easily b e seen that con v ( F ′ ∪ { v } ) is not totally splittable for d > 3 sine F ′ do es not ha v e an y splits. If F ′ is a prism o v er a simplex, w e are done b y indution. An easy onsideration of the ases, whi h w e omit, allo ws us to pro v e the result in the base ase d = 3 . See Example 20 and Figure 3 for one of the ases arising.  Example 20. Consider the 3 -p olytop e P = conv { e 1 , e 2 , e 3 , − e 2 − e 3 , − e 1 − e 3 , − e 1 − e 2 } , whi h is a prism o v er a triangle. F or instane, the p oin t v = e 1 + e 2 − e 3 lies almost b ey ond the quadrangular faet F = con v { e 1 , e 2 , − e 2 − e 3 , − e 1 − e 3 } . The p olytop e con v ( P ∪ { v } ) admits a triangulation whi h is not split; see Figure 3. Prop osition 21. L et P b e a pr op er total ly splittable p olytop e that is not a r e gular r ossp oly- top e. Then P is a join if and only if the vertex set of P admits a p artition V ert P = U ∪ W suh that no vertex split of a vertex in U is  omp atible with any vertex split of a vertex in W . Pr o of. Let P = (conv U ) ∗ (conv W ) b e a prop er join. In partiular, P is not a p yramid, and con v U and con v W b oth are at least one-dimensional. Then, b y the denition of join, ea h v ertex in U shares an edge with ea h v ertex in W , and th us the orresp onding v ertex splits are not ompatible. Con v ersely , assume that no split with resp et to a v ertex in U is ompatible with a split with resp et to an y v ertex in W . By Lemma 7 ea h v ertex in U is joined b y an edge to ea h v ertex in W . Prop osition 12 sa ys that ea h v ertex split h yp erplane u ⊥ on tains all neigh b ors of u . Th us w e infer that T u ∈ U u ⊥ ⊃ con v W and, symmetrially , T w ∈ W w ⊥ ⊃ conv U . No w there are t w o ases to distinguish. If T v ∈ V ert P v ⊥ is non-empt y then P is a regular rossp olytop e due to Prop osition 17 on traditing the assumption. TOT ALL Y SPLITT ABLE POL YTOPES 11 The remaining p ossibilit y is that T v ∈ V ert P v ⊥ is empt y . In this ase w e ha v e aff U ∩ aff W ⊆ \ w ∈ W w ⊥ ∩ \ u ∈ U u ⊥ = \ v ∈ V ert P v ⊥ = ∅ . The ane subspaes aff U and aff W are sk ew. It follo ws that P = (conv U ) ∗ (conv W ) .  F or the follo wing w e will swit h from the primal view on our p olytop e P to its spherial Gale dual G . A p oin t of m ultipliit y t w o in G is alled a double p oint . V erties of P orresp onding to the same p oin t in G are alled siblings . Lemma 22. L et P b e a total ly splittable p olytop e whih is not a join, and let G b e a spheri al Gale diagr am of P . Then P is pr op er, and e ah p oint of G is a single p oint, or e ah p oint is a double p oint. In p artiular, ther e ar e no p oints in G with multipliity gr e ater than two. Pr o of. If P is a regular rossp olytop e w e kno w from the expliit desription of G in Example 9 (iii ) that the onlusion of the lemma holds. So w e an assume that this is not the ase. Sine w e assume that P is not a join, in partiular, it is not a p yramid, and this is wh y P is prop er. If G had a p oin t with m ultipliit y three or ab o v e, then ea h pair of opies of x denes a iruit whi h yields a on tradition to Lemma 5. So supp ose no w that v 1 is a v ertex that has a sibling v 2 and that the set W of all v erties without a sibling is non-empt y . Then, again b y Lemma 5, v ⋆ 1 = v ⋆ 2 is not on tained in p os W ⋆ . By the Separation Theorem [7, 2.2.2℄, there is an ane h yp erplane in R n − d − 1 whi h w eakly separates v ⋆ 1 = v ⋆ 2 from p os W ⋆ . This argumen t ev en w orks for all v erties with a sibling sim ultaneously . That is H w eakly separates the double p oin ts from non- double p oin ts. By rotating H sligh tly , if neessary , w e an further assume that H on tains at least one dual v ertex w ⋆ of a v ertex w ∈ W without a sibling. F or ea h su h w ∈ W with w ⋆ ∈ H the supp ort of the iruit C [ w ] is a subset of W ⋆ and from Lemma 5 it follo ws that the supp ort of C [ w ] is on tained in the h yp erplane H . In the primal view, this means that all v erties v of P with v ⋆ 6∈ H ha v e to b e in the splitting h yp erplane w ⊥ and that the v ertex split of w annot b e ompatible to an y v ertex split of a v ertex v with v ⋆ 6∈ H . If no w w e dene U := { w ∈ V ert P | w ⋆ ∈ H } w e ha v e a partition of V ert P in U and V ert P \ U su h that no v ertex split of a v ertex in U is ompatible with an y v ertex split of a v ertex in V ert P \ U . So P is a join b y Prop osition 21 .  1 5 3 7 2 6 4 8 Figure 4. Gale diagram of the join of t w o squares, lab eled { 1 , 2 , 3 , 4 } and { 5 , 6 , 7 , 8 } , resp etiv ely . 12 HERRMANN AND JOSWIG A p oin t x ∈ G is antip o dal if − x is also in G . Notie that an y quadrangle, regular or not, has a zero-dimensional spherial Gale diagram with exatly t w o pairs of an tip o dal p oin ts. Lemma 23. L et P b e a total ly splittable d -p olytop e with d ≥ 2 whih is not a join. If e ah p oint in the spheri al Gale diagr am G is a double p oint then P is a r e gular r ossp olytop e. Pr o of. Assume that ea h p oin t in G is a double p oin t. Let v b e an y v ertex of P and v ⊥ the h yp erplane orresp onding to the v ertex split of v . Sine v ⋆ is a double p oin t in G there is exatly one v ertex w other than v whi h is not on tained in v ⊥ . The p olytop e Q := con v (V ert P \ { v , w } ) = P ∩ v ⊥ is a fae of the v ertex gure of v and hene totally splittable b y Prop osition 12 . Clearly , a spherial Gale diagram of Q again has only double p oin ts. Indutiv ely , w e an th us assume that Q is a regular rossp olytop e. Therefore, its split h yp erplanes ha v e a non-empt y in tersetion. Sine this in tersetion is on tained in v ⊥ it follo ws that the split h yp erplanes of P also ha v e a non-empt y in tersetion. Hene P is a regular rossp olytop e b y Prop osition 17. As a basis of the indution w e an onsider the ase where G is on tained in S 1 . As G m ust span R 2 , and as ea h p oin t in G o urs t wie, the p olytop e P has six v erties, and it is three-dimensional. It an b e sho wn that P is a regular o tahedron. The t w o-dimensional ase will b e dealt with in the pro of of Lemma 25 b elo w.  Lemma 24. L et P b e a total ly splittable d -p olytop e with d ≥ 2 whih is not a join. If e ah p oint in the spheri al Gale diagr am G is antip o dal then P is a prism over a simplex. Pr o of. Supp ose that ea h p oin t in G is an tip o dal. Let k := n − d − 1 b e the dimension of the linear span of G , and so w e an view G as a subset of S k − 1 . W e laim that the n um b er of v erties of P equals n = 2 d or, equiv alen tly , that n = 2 k + 2 . Pi k an y o iruit of G . This orresp onds to a linear h yp erplane H in R k whi h on tains at least 2 k − 2 p oin ts of G , due to an tip o dalit y . Sine G is the Gale diagram of a p olytop e ea h op en halfspae dened b y H on tains at least 2 p oin ts [12 , Theorem 6.19℄. W e onlude that n ≥ 2 k + 2 . No w w e will sho w that n ≤ 2 k + 2 hene n = 2 k + 2 . T o arriv e at a on tradition, supp ose that the spherial Gale diagram G on tains at least k + 2 an tip o dal pairs. T ak e an y v ertex v of P , and let v ⋆ b e its dual in G . Pi k an ane h yp erplane H ⋆ in R k whi h is orthogonal to v ⋆ and su h that v ⋆ and the origin are on dieren t sides of H ⋆ . Let W = { v ⋆ 1 , v ⋆ 2 , . . . , v ⋆ m } b e the set of p oin ts in G distint from v ⋆ for whi h the orresp onding ra ys in terset H ⋆ . Firstly , m ≥ k + 1 sine G on tains k + 1 an tip o dal pairs in addition to v ∗ and its an tip o de. Seondly , v ⋆ is in the p ositiv e span of the ra ys orresp onding to the p oin ts in W sine among those p oin ts are the elemen ts of C [ v ] + . By Carathéo dory's Theorem [7, 2.3.5℄ w e an assume that the orresp onding ra ys of v ⋆ 1 , v ⋆ 2 , . . . , v ⋆ k +1 still on tain v ∗ in their p ositiv e span. Let Q b e the on v ex h ull of the in tersetions of the ra ys orresp onding to v ⋆ 1 , v ⋆ 2 , . . . , v ⋆ k +1 with the h yp erplane H ⋆ . No w Q is a ( k − 1) -dimensional p olytop e with k + 1 v erties. Su h a p olytop e has preisely t w o triangulations ∆ and ∆ ′ ; these are related b y a ip , see [5, 2.4.1℄. Let σ and σ ′ b e maximal simplies of ∆ and ∆ ′ on taining the p oin t ( R v ⋆ ) ∩ H ⋆ . By onstrution σ giv es rise to a iruit D of G whose negativ e supp ort orresp onds to the v erties of σ and its p ositiv e supp ort orresp onds to v ⋆ . Similarly , σ ′ denes another su h iruit D ′ . Sine no maximal simplex of ∆ also o urs as a maximal simplex in ∆ ′ w e ha v e σ 6 = σ ′ implying D 6 = D ′ . This on tradits Lemma 5, and this nally pro v es that n equals 2 k + 2 . By no w w e kno w that G onsists of preisely k + 1 an tip o dal pairs in S k − 1 . So P is a d -p olytop e with 2 k + 2 = 2 d v erties. W e ha v e to sho w that P has the same orien ted matroid as a prism o v er a ( d − 1) -simplex. This will b e done b y sho wing that the o iruits of G (whi h are the iruits of P ) agree with the iruits of a prism o v er a simplex. So TOT ALL Y SPLITT ABLE POL YTOPES 13 onsider the prism o v er a simplex with o ordinates as in the pro of of Lemma 19. Then ea h iruit C of this prism is of the form C + = { e i , f j } and C − = { f i , e j } for distint i and j . Moreo v er, e ⋆ i and f ⋆ i are an tip o des in the prism's spherial Gale diagram. The o iruits of G are giv en b y all (linear) h yp erplanes in R k spanned b y k − 1 pairs of p oin ts in G . None of the other t w o pairs of p oin ts an b e on tained in su h a h yp erplane sine G is the Gale diagram of a p olytop e [12 , Theorem 6.19℄. So the o iruits of G are giv en b y C ⋆ + = { x, y } , C ⋆ − = {− x, − y } for all distint x, y ∈ G with x 6 = ± y .  Lemma 25. L et P b e a total ly splittable d -p olytop e with d ≥ 2 whih is not a join. If e ah p oint in the spheri al Gale diagr am G is b oth a double p oint and antip o dal then d = 2 , and P is a quadr angle. Pr o of. If ea h p oin t in G is an tip o dal from Lemma 24 w e kno w that P is a prism o v er a ( d − 1) -simplex. The only ase in whi h su h a Gale diagram has the prop ert y that ea h p oin t is a double p oin t is d = 2 , and P is a quadrangle.  No w w e ha v e all ingredien ts to pro v e our main result. Pr o of of The or em 8. Let P b e a totally splittable d -p olytop e with spherial Gale dual G . By Lemma 16 , w e an assume without loss of generalit y that P is not a join. Consider a v ertex v ∈ V ert P with the prop ert y that v ⋆ is neither a double nor an an tip o dal p oin t. By Prop osition 12 , the p olytop e Q := con v (V ert P \ { v } ) obtained from P b y the deletion of v is again totally splittable. Moreo v er, dim Q = d sine P is not a p yramid. Let us assume for the momen t that Q is also not a join. Then w e an rep eat this pro edure un til after nitely man y steps w e arriv e at a p olytop e P ′ with a spherial Gale diagram G ′ whi h onsists only of double and an tip o dal p oin ts. In this situation Lemma 22 implies that all p oin ts of G ′ are double p oin ts or all p oin ts of G ′ are an tip o dal. Com bining Lemma 23 , Lemma 24 , and Lemma 25 , w e an onlude that either d = dim P = dim P ′ = 2 and P ′ is a quadrangle, or d ≥ 3 and P ′ is a regular rossp olytop e, or d ≥ 3 and P ′ is a prism o v er a simplex. The question remaining is whether P and P ′ an atually b e dieren t. F or d ≥ 3 this is ruled out b y Lemma 18 (if P ′ is a rossp olytop e) and Lemma 19 (if P ′ is a prism). In the nal ase dim P = dim Q = dim P ′ = 2 . The pro of of our main result will b e onluded with the subsequen t prop osition.  Prop osition 26. L et P b e a total ly splittable p olytop e with spheri al Gale diagr am G , and let v b e a vertex of P with the pr op erty that its dual v ⋆ in G is neither a double nor an antip o dal p oint. If P is not a join then neither is Q := con v (V ert P \ { v } ) . Pr o of. By [3, Lemma 3.4℄, the Gale transform of Q is the minor G/v ⋆ obtained b y on- trating v ⋆ in G . Up to an ane transformation w e an assume that v ⋆ is the rst unit v etor in R n − d − 1 , and so G/v ⋆ is the pro jetion of G \ { v ⋆ } to the last n − d − 2 o ordinates. W e all the pro jetion map π . Sine v ⋆ is neither an tip o dal nor a double p oin t, no p oin t in G/v ⋆ is a lo op, and th us Q is prop er, that is, it is not a p yramid. So supp ose that Q = Q 1 ∗ Q 2 is a join with dim Q 1 ≥ 1 and dim Q 2 ≥ 1 . Then there are spherial Gale diagrams G 1 and G 2 of Q 1 and Q 2 , resp etiv ely , su h that G/v ⋆ = G 1 ⊔ G 2 as a m ultiset in S n − d − 3 . Up to ex hanging the roles of Q 1 and Q 2 , there is a faet F 1 of Q 1 su h that v ⊥ ∩ P , whi h is a faet of Q , is a join F 1 ∗ Q 2 . That is to sa y , the osupp ort of the iruit C [ v ] , orresp onding to the v ertex split of v in P , is mapp ed to G 1 b y π . In partiular, v ⋆ is not in the p ositiv e h ull of the p oin ts dual to the v erties of Q 2 . 14 HERRMANN AND JOSWIG The Separation Theorem [7, 2.2.2℄ implies that there is a linear h yp erplane H in R n − d − 1 separating v ⋆ from the duals of the v erties of Q 2 . As in the pro of of Lemma 22 w e an no w argue that P is a join, whi h on tradits our assumptions.  This nally ompletes the pro of of the theorem. Remark 27. If v ⋆ is an tip o dal or a double p oin t, then Q is a p yramid o v er the unique faet of Q whi h is not a faet of P . This sho ws that the assumption on v ⋆ in Prop osition 26 is neessary . F or instane, b y insp eting the t w o Gale diagrams in Figure 2 one an see diretly that if P is a regular o tahedron or a prism o v er a triangle, in b oth ases Q is a p yramid o v er a quadrangle. Remark 28. A triangulation ∆ of a d -p olytop e is foldable if the dual graph of ∆ is bipartite. This is equiv alen t to the prop ert y that the 1 -sk eleton of ∆ is ( d + 1) -olorable. In [9, Corollary 4.12℄ it w as pro v ed that an y triangulation generated b y splits is foldable. This means that ea h triangulation of a totally split p olytop e is neessarily foldable. W e are indebted to Raman San y al for sharing the follo wing observ ation with us. Corollary 29. Eah total ly splittable p olytop e is e quide  omp osable. A p olytop e is e quide  omp osable if ea h triangulation has the same f -v etor. Pr o of. This follo ws from the lassiation ase b y ase: Ea h triangulation of an n -gon has exatly n − 2 triangles. Ea h triangulation of a d -dimensional regular rossp olytop e has exatly 2 d − 2 maximal ells. Ea h triangulation of a prism o v er a ( d − 1) -simplex has exatly d maximal ells. A similar oun t an b e done for the lo w er dimensional ells. Observ e that equideomp osabilit y is preserv ed under taking joins.  It w ould b e in teresting to kno w if Corollary 29 has a diret pro of without relying on Theorem 8. Remark 30. Ba y er [2℄ denes a p olytop e to b e we akly neighb orly if an y k of its v erties are on tained in some fae of dimension 2 k − 1 . She sho ws that a w eakly neigh b orly p olytop e is neessarily equideomp osable [2, Corollary 10℄. Prisms o v er simplies are w eakly neigh b orly whereas rossp olytop es are not; so the approa h of Ba y er is somewhat transv erse to ours. Moreo v er, all iruits of a totally splittable p olytop e are b alan e d in the sense that the p ositiv e and the negativ e supp ort share the same ardinalit y . 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Geom. 36 (2006), no. 2, 331361. MR MR2252108 (2007f:52025) 11. Ido Shemer, Neighb orly p olytop es , Israel J. Math. 43 (1982), no. 4, 291314. MR MR693351 (84k:52008) 12. Gün ter M. Ziegler, L e tur es on p olytop es , Graduate T exts in Mathematis, v ol. 152, Springer-V erlag, New Y ork, 1995. MR MR1311028 (96a:52011) Sven Herrmann, F a hbereih Ma thema tik, TU D armst adt, 64289 D armst adt, Germany E-mail addr ess : sherrmannmathema ti k. tu- da rm sta dt .d e Mihael Joswig, F a hbereih Ma thema tik, TU D armst adt, 64289 D armst adt, and Insti- tut für Ma thema tik, TU Berlin, 10623 Berlin, Germany E-mail addr ess : joswigmath.tu-be rl in .de