Asymptotics of Convex sets in En and Hn

ASYMPTOTICS OF CONVEX SETS IN E n AND H n . IGOR RIVIN A bstract . W e study convex sets C o f finite (but non-zero) volume in H n and E n . W e show that the intersection C ∞ of any such set with the ideal boundary of H n has M inkowski ( and thus Hausdor ff ) dimension of at m ost ( n − 1 ) / 2 . and this bound is sharp, at least in some dimensions n . W e also show a sharp bound when C ∞ is a smoo th submanifold of ∂ ∞ H n . In the hyperbolic ca se, we show that for any k ≤ ( n − 1) / 2 there is a bounded sec tion S of C through any prescribed p , and we show an upper bound on the rad ius of the ball ce ntered at p containing such a section. W e show similar bounds for se c tions through the origin of a convex body in E n , and give asymptotic estimates as 1 ≪ k ≪ n . . I ntroduction The work in this note was motivated by a question of Itai Benjamini and Nir A vni on whether there is any version of A. Dvoret sky’s The- or em valid in high-dimensional hyperbolic spaces H n . It quickly became apparent that in or der t o have any hope of answering this question one must have a good underst a nd in g of the geometry of convex sets in H n at (and near) the ideal bo undary , and this is the subject of this work. The most basic q uestion of this type is to un- derstand the geometry of the “ id e al part” C ∞ of a convex set C with nonempty interior and finite volume (a simpler way of putting it is r eq uiring 0 < V ( C ) < ∞ . ) The first most basic question is: what is the dimension of C ∞ ? Ther e ar e, of cour se, many definitions of dimen- sion, but the most natural one for our purposes turns out to be the (upper) Minkowski dimension dim M . Using a simple geometric idea we show that dim M ( C ∞ ) ≤ n − 1 2 , Date : November 21, 2018. Key words and phrases. hyperbolic, volume, dimension. The a uthor would like to thank Stanford University f or their hospitality during the preparation of th is wo rk. He would also like to thank Itai Benjamini for the question that led to this line of inquiry , Jean-Ma rc Schlenker for e nlightening correspondence, and to Peter Storm for helpful conversa tions. 1 2 IGOR RIVIN Since the Minkowski dimensions are both upp e r bounds on the H a us- dor ff dimension, we have the same bound on the Ha usdor ff dimen- sion. W e show furt her that for C ∞ smooth, the volume of the convex hull of C ∞ is finite whenever the (to p ological) dimension of C ∞ is not gr eater than ⌊ n / 2 ⌋ − 1 , and that bound is sharp. In dimension 3 , we show that ther e are sets C ∞ , of arbitrary Haus- dor ff dime nsion smaller than 1 , such that the volume of the convex hull of C ∞ is finite. W e do not know whether ther e are sets of Haus- dor ff dimension equal to 1 with that pro p e rty . The next question is whe ther there is always a k -dimensional pla ne thr ough any fixed point p of C of bo unded di a m e ter . The dimen- sion e stimate above essentially shows that the answer is a ffi rmative (whenever k does not exceed the critical dimension ( n − 1) / 2, but with so me e xtra work we can show more pr ecise estimates on e x- actly how small we can get the diameter in terms of n , k , V ( C ) and the “thickness” of C (that is, the radius of the largest ball center ed on p and contained in C . ) The nature of the ar gument is such that we can, essentially without change, obtain estimates of the sort ”intersections of at least 30% of all planes thr ough p are contained in B ( p , r ) . The basic idea is simple: if we let Ω r ( C ) be the set of dir ections in which rays of length r em a nating fr om p a re contained in C , then in or der to estimate the measur e of the set of planes which intersect Ω r ( C ) we pro d uce a bound on the measure of the ǫ -neighbor hood of Ω r ( C ) (which is always a B orel set, unlike Ω r ( C ) itself). T o pr oduce such a bo und we use a couple of simple geometric ideas, the first (trivial) one giving a bound on Ω r ( C ) as a function of r , an d secondly , using the Double Cone Lemma (in Section 4) we show that a certain ǫ -neighbor hood of Ω r ( C ) is contained in an Ω s ( C ) , for some s ( r , ǫ ) . The outline of the paper is as follows: In Section 1 we recall the basic definitions of Minkowski mea sure and content. In Section 2 we recall so me basic formulas and estimates on the volumes of balls in H n . In Section 3 we recall some of the p r operties of the Klein model of H n . In Section 4 we describe our basic geometric tool – the “Double Cone Lemma.” In Section 5 we pro ve the basic estimates on the limit sets of finite volume convex sets in H n . Our main result s are Theor em 5.2, which states that the upper Mink owski dimension (hence the Hausdor ff dimension) of the limit set of a convex set of finite volume in H n is bounded above by ( n − 1) / 2 , and Theor em 5.4, which observ es ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 3 that the volu me of a convex hull of a smooth subset S of the ideal boundary of H n is finite if and only if the (topological) dimension of S is no greater than ⌊ n / 2 − 1 ⌋ . In Section 9 we construct a family of sets in ∂ H 3 of Hausdor ff di- mension tending to 1 , such that the volume of the hyperbolic convex hull of each of the se t is finite, showing that the r esult of Section 5 ar e sharp (at least in dimension 3). In Section 6 we study the sizes of the intersections of a non- degenerate convex set C ∈ H n with planes thr ough a fixed point p . The main r esult is Theor em 6.3, which is too cumbersome to state here, but implies (for large n and C of lar ge volume V ( C )) that one can find such a section of dimension 1 ≪ k ≪ n cont a ine d in a ball of radius about 1 2 log n bigger than the radius of a rou nd ball of volume V ( C ) in H n . In Section 7 we apply our methods to similar questions in Eu- clidean space, where, not surprisingly , the estimates come out quite di ff erently . The main result is Theorem 7.5, which implies that for a 1 ≪ k ≪ n , ther e is a section of C contained in a ball of radius about √ n / 2 π e bigger than the radius of a round ball in E n of volume V ( C ) . In Section 8 we p rove the basic technical estimates we need. 0.1. Notation. W e shall denote the volume of a ball of radius r in E n , S n , H n by V n E ( r ) , V n S ( r ) , V n H ( r ) , respectively . In addition we will use the notation Λ n X ( V ) for the inverse function of V n X , for X = E , H , S – that is, Λ n X ( V n X ( r )) = r . W e will also use the standard notation κ n = V n E (1) , and also ω n for the area of the spher e of unit radius in E n . As in the previou s sentence, we will use X when the statement do e s not depend on which of the thr ee a mbient spaces we are talking about. W e will frequently use the following function: Definition 0.1. Let r 1 > r 2 > r 0 . Then, we defi ne α r 0 ( r 1 , r 2 ) = asin            r 0 q r 2 1 − r 2 0 − q r 2 2 − r 2 0 r 2            . W e will denote the ǫ ne ighbor hood of a subset S of S k by S ǫ . In some places below we use the notation µ ( S ) for subsets of S k not assumed Lebesgue measurable. In such cases µ stands for the lower Minkowski content of S , namely µ ( S ) = lim inf ǫ → 0 λ ( S ǫ ) , wher e λ is Lebesgue mea sure. W e will also use the no tation ν ( S ) for the normalized p robability measur e o f S (in other wo rds, ν ( S ) = 4 IGOR RIVIN µ ( S ) /ω k + 1 , so that ν ( S k ) = 1 . For discus sion o f Minkowski content (and all other measure-theor etic concepts), the reader is r eferred to P . Mattila’s book [6]. 1. M inkowsky measure and dimension This section is shamelessly stolen from P . Mattila’s book [6]; we include it here in an attempt to keep this paper self-contained. The setup is as follows: Let A be a non-empty bounded subset of R n or S n . Denoting the n -dimensional Lebesgue measure by λ, as befor e, we define the upper s -dimensional Minkowski content of A by M ∗ s = lim sup ǫ → 0 (2 ǫ ) s − n λ ( A ǫ ) , and the lower s -dimensional Minkowski content by M s ∗ = lim inf ǫ → 0 (2 ǫ ) s − n λ ( A ǫ ) . Using these, we can define the upper Minkowski dimension as: dim M A = inf { s : M ∗ s ( A ) = 0 } = sup { s : M ∗ s ( A ) > 0 } . Similarly , the lower Minkowski dimension is: dim M A = inf { s : M ∗ s ( A ) = 0 } = sup { s : M ∗ s ( A ) > 0 } . 2. G eometr y of B alls and S pheres Recall that: κ n = π n / 2 Γ ( n / 2 + 1) , (1) ω n = n κ n . (2) The following is also classical: Theorem 2.1. V n E ( r ) = ω n Z r 0 r n − 1 dx = κ n r n , V n H ( r ) = ω n Z r 0 sinh n − 1 dx , V n S ( r ) = ω n Z r 0 sin n − 1 dx . ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 5 Lemma 2.2. Let Ω ⊆ S n − 1 ⊂ E n = T p ( X ) , and let C Ω ( r ) be the cone over Ω , that is, the convex hull of r Ω and the origin under the e xponential map (in particular , if Ω = S n − 1 , C Ω ( r ) is j ust the ball of radius r around p . The, the volume of C Ω ( r ) satisfies V ( C Ω ( r )) ≥ µ ( Ω ) V n X ( r ) = ν ( Ω ) ω n V n ( r ) , with equality if Ω ( r ) is Lebesgue-measurable Proo f . The statement is immediate for Lebesgue-measurable sets, and for general sets the inequality is a d irect consequence of the definition of the Minkowski content µ.  Corollary 2.3. Let C be a convex body in X , and p ∈ C . Let Ω R ( C ) be the set of those unit θ in the unit tangent s phere at p for which the exponential map of the seg m ent fro m the origin to R θ is contained in K . Then, µ ( Ω R ( C )) ≤ V ( C ) / V n X ( r ) and thus ν ( Ω R ( C )) ≤ V ( C ) /ω n V n X ( r ) . Proo f . The cone of radius R over Ω R ( C ) is cont ai ne d in C , so the estimate of Lemma 2.3 applie s.  2.1. V olume asymptotics. Lemma 2.4. As r goes to infinity , V n H ( r ) is asymptotic to ω n e ( n − 1) r 2 n − 1 ( n − 1) ; for all r > log 2 / 2 , ω n e ( n − 1) r 2 n − 1 ( n − 1) > V n H ( r ) > ω n e ( n − 1) r 4 n − 1 ( n − 1) . Proo f . Immediate from Theor em 2.1.  Remark 2.5 . Lemma 2.4 can be thought of stating that a ball in H n of lar ge volume V has radius r = Λ n H ( V ) ∼ log 2 n − 1 ( n − 1) V ω n ! n − 1 . For n ≫ 1 , Stirling’s formula tells us that (3) r ∼ log 2 / 2 − log π/ 2 − 1 / 2 + log n / 2 + log ( V ) / ( n − 1) . 6 IGOR RIVIN 3. T he K lein model of H n The Klein Model K : H n → B n (0 , 1) is a repr esentation of H n as the interior of the unit ball in E n . I t has the virtue that it is geodesic , so that the images of totally geodesic subspaces of H n ar e intersections of a ffi ne subspaces of E n with B n (0 , 1) . Consequently , the images of convex sets of H n under K are also convex. The hyperbolic metric can be recover ed from B n (0 , 1) as follows: If p , q ∈ B n (0 , 1) , then, denoting the hyperbo lic distance between K − 1 ( p ) , K − 1 ( q ) by d H ( p , q ) , we have the formula 1 d H ( p , q ) = arccosh       1 − p · q p 1 − p · p p 1 − q · q       . In particular , if p 0 = K − 1 ( 0 ) , then (4) d H ( p 0 , q ) = 1 2 log 1 + k q k 1 − k q k ! . Conversely , if p , q ∈ H n and K ( p ) = 0 , and d ( p , q ) = R , then (5) k K ( q ) k = tanh R . The hyperbolic metric can be e xpr essed (see, eg, [8]) as follows in the Klein model. Fir st, we use polar coordinates: d x = dr 2 + r 2 k d u k 2 . Hyperbolic metric is then written as: ds 2 = dr 2 (1 − r 2 ) 2 + r 2 k d u k 2 1 − r 2 , showing that K − 1 at q distorts distances by a factor of 1 p 1 / k q k 2 − 1 , in the spherical direction, but by a factor of 1 1 − k q k 2 radially . 1 A geometric way to understand the below formula is a s Hilbert distance on the ball – if the line through p , q intersects the unit sphere a t u , v , then d H ( p , q ) = 1 / 2 log ( [ u , p , q , v ]) , where [] denotes the cross-ratio. ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 7 Finally , if we define Ω K d ( C ) to be the set on the visual spher e of 0 of the points of a convex body C outside the (Euclidean) ball of radius d ≫ 1 , then the formula (4) together with Corollar y 2.3 tell us: Lemma 3.1. With definitions as above, µ ( Ω K d ( C )) ≤ ( n − 1)2 n − 1 V ( C ) ω n 1 − d 1 + d ! n − 1 2 ≤ ( n − 1)2 n − 1 2 V ( C )(1 − d ) n − 1 2 ω n . 4. T he double cone W e will be using the following construction: Suppose 0 < r 0 < r 1 < r 2 , and let C be a closed co nvex subset of B n (0 , r 2 ) . A ssume further that B n (0 , r 0 ⊂ C ) , and that C r 2 = C ∪ ∂ B n (0 , r 2 ) , ∅ . Consider now ξ ∈ C r 2 , and the ball B n (0 , r 1 ) . and the cone H ( ξ , r 0 ) , which is the convex hull of B (0 , r 0 ) and ξ. The cone H ( ξ , r 0 ) intersects ∂ B n (0 , r 1) in a disk D ( ξ, r 1 , r 0 ) , and we have the following: Lemma 4.1. The dis k D ( ξ, r 1 , r 0 ) has angular radius α r 0 ( r 2 , r 1 ) . Proo f . By ro tational symmetry , it su ffi ces to consider the pla na r case ( n = 2). Let the two tangents to ∂ B 2 (0 , r 0 ) fr om ξ be l 1 and l 2 , and let l i ∩ ∂ B 2 (0 , r 0 ) be t 1 and t 2 , respectively . By the Pythagor ean theor em , | ξ t 1 | = q r 2 2 − r 2 0 . Le t now s 1 = l 1 ∩ ∂ B 2 (0 , r 1 ) , and let p be the base of the perpendicular dr opped from s 1 onto O ξ. The triangle ξ s 1 p is similar to the triangle ξ Ot 1 , and since | ξ t 1 | = q r 2 1 − r 2 0 , it follows that | ps 1 | = r 0  q r 2 2 − r 2 0 − q r 2 1 − r 2 0  , and the assertion of the le mma follows immediately .  which in turn implies that Ω K d contains a (spherical) disk D ξ ( α r 0 ( d )) of radius α r 0 ( d ) aro und ξ. Theorem 4.2. With notation as above, let Ω r be the s e t of rays from the origin to C r = C ∩ ∂ B n (0 , r ) , (identified with a subset of the “visual s phere” at the origin – the unit tangent sphere). Then, the α r 0 ( r 2 , r 1 ) neighborhood of Ω r 2 is contained in Ω r 1 . Proo f . Consider a point η ∈ Ω r 2 . By Lemma 4.1, the cone J α r 0 ( r 2 , r 1 ) (0) with the vertex at the origin and angle α r 0 ( r 2 , r 1 ) is contained in Ω r 1 , which is precisely the statement of the Corollary .  Remark 4.3 . The cones H and J give this se ction its name. 8 IGOR RIVIN 5. A pplica tio ns to limit sets Let C be a convex body in H n . W e will say that the limit set of C – denoted by C ∞ – is the intersection of (the closur e of) C with the ideal boundary of H n . In the Klein model, K ( C ∞ ) = K ( C ) ∩ ∂ B n (0 , 1) . Note that in the Klein model we can ide ntify the ideal boundary of H n with the unit tangent spher e a t the origion. W ith that identification, using the notation of Corollary 2.3, we can d e fine C ∞ = \ R Ω R ( C ) . = \ d Ω K d ( C ) . In t he sequel, we will assume that C has non-empty interior , and fr om now on, all computations will be in the Klein model. W e then assume particular , ther e is a ba ll B 0 of radius r 0 center ed on the origin and contained in K ( C ) . Assume now that C has finite volume V ( C ) . Theor em 4.2 allows us to str engthen Lemma 3.1 as follows: Lemma 5.1. µ (( Ω K d 1 ( C )) α r 0 ( d 1 , d 2 ) ) ≤ ( n − 1)2 n − 1 V ( C ) ω n 1 − d 2 1 + d 2 ! n − 1 2 ≤ ( n − 1)2 n − 1 2 V ( C )(1 − d 2 ) n − 1 2 ω n . W e are now ready to show: Theorem 5.2. Let C be a convex set in H n of finite volume with nonempty interior . Then, the (upper) Mi nkowski dimension of C ∞ is at most ( n − 1) / 2 . Proo f . Set d 1 = 1 in the statement of Lemma 5.1. By Lemma 8.2 we see that Ω K 1 ( C ) asin( r 0 (1 − d )) ⊆ Ω K d ( C ) asin r 0 d 1 − d ⊆ ( Ω K d ( C )) α 1 , d , and so (setting ǫ = asin( r 0 (1 − d ))) µ  Ω K 1 ( C ) ǫ  ≤ ( n − 1)2 n − 1 2 V ( C ) sin n − 1 2 ǫ r n − 1 2 0 ω n . Letting ǫ tend to 0 , we see that the measur e of Ω K 1 ( C ) ǫ is bounded above by a constant times ǫ n − 1 2 , whence the result.  ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 9 Corollary 5.3. With C as above, the Hausdor ff dimension of C ∞ is at most ( n − 1) / 2 . Proo f . The Minkowski dimensions (upper and lower) ar e both upper bounds on the Hausdor ff dime nsion.  5.1. Are the r e sults on dim e nsion sharp? As observed by Peter Storm, the bound of The orem 5.2 is clea rly not sharp in dimension 2 . Ther e , since the ar ea of an ideal triangle is always π, it is ea sy to see that if C has finite area, C ∞ is a finite set, a nd hence a ny reasonable dimension of C ∞ equals 0 . On the other hand, Eq. (3) indicates that the hyperbolic volume element at q is propor tional to the Euclidean volume e lement divided by r ( n + 1) / 2 . This indicates that the conv e x hull C of a (very) small totally geodesic disk D k (of dimension k ) on ∂ B n (0 , 1) a nd a ball F in the interior of B (0 , 1) looks, near the ideal boundary , as a Cartesian pr oduct of D and the cone from a point p ∈ D onto a section F ⊥ of F orthogo nal to D . Since the dimension of F ⊥ equals n − k , we see that we have shown: Theorem 5.4. Let M be a piecewise smooth embedded s ubmanifold of ∂ B n (0 , 1) of dime nsion d , and le t C ( M ) be the convex hull of M . Then the hyperboli c volume of C ( M ) i s finite if and only if k is smaller — than n − ( n + 1) / 2 = ( n − 1) / 2 . Remark 5.5 . The regularity r equired in the statement of Theor e m 5.4 is not very oner ous: C 1 is certainly su ffi cient; presumably rectifi a ble is also. Theor em 5.4 indicates that the bound of Theor em 5.2 is sharp for piecewise smoot h sets and when n is even. For arbit rary sets, we show a lower bound (at le a st when n is 3) in Section 9. Remark 5.6 . By the result s of B. Colbois and P . V erovic [1], the r esults of this note apply essentially without change to convex bodies in the Hilbert metric on arbitrary smooth convex domains. 6. A pplica tio ns to central sections In this sect ion we will apply the above r esults to the following question: Suppose we have a convex set C with nonempty interio r and finite volume V ( C ) in X , and a point p ∈ C . For each k -dimensional plane Π throug h p , consider Π C = Π ∩ C , and let d ( Π ) = ma x x ∈ Π C ( d ( x , p ) – in other wor ds, 10 IGOR RIVIN d ( Π ) (not ne ce ssarily finite) is the radius of the smallest spher e containing Π C . The question, then, is do we ha ve any upper bound on the smallest d ( Π )? In this form, the que stion is not hard to answer using our r esults above. First, we will need the follo wing standard fact (see, eg, [7, page 4]): Theorem 6.1. Let 1 ≤ k ≤ n , le t ζ ∈ G n , k , where G n , k is the Grassmannian of k planes through the origin in R n . D enote by S ( ζ ) = S n − 1 ∩ ζ the unit sphere of ζ. Then Z S n − 1 f d ν = Z G n , k Z S ( ζ ) f ( t ) d ν ζ ( t ) d ν ( ζ ) for all f ∈ L 1 ( S n − 1 ) , where ν ζ is the normalized Haar measur e on S ( ζ ) , ν on the left is the normalized Haar measure on S n − 1 and on the r i ght on G n , k . W e will be applying The orem 6.1 to the indicator function f ǫ ( M ) of the ǫ -neighborhoo d M ǫ of a set M ⊆ S n − 1 . Every k -plane which intersects M intersects M ǫ in at least an ǫ -ball, a nd therefor e if ever y k -plane intersects K , we have the inequality: (6) µ ( M ǫ ) ≥ V k S ( ǫ ) , which implies: Theorem 6.2. There is a k-plane Π k throu gh the origin such that ( Π k ∩ C ) ⊆ B n 0 , 1 2 log 1 + d 1 − d ! if µ (( Ω K d ) ǫ ) V k − 1 S ( ǫ ) < 1 for some ǫ > 0 . Now , let M = Ω K d ( C ) . where C satisfies the hypotheses of the be- ginning of this note (in particular , contains a ball of radius r 0 ar ound the origin). By Lemma 5.1, for any d 2 < d , we have ν (( Ω K d ) α r 0 ( d , d 2 ) ) ≥ ( n − 1)2 n − 1 2 V ( C )(1 − d 2 ) n − 1 2 ω 2 n . W e assume in the sequel that V ( C ) is large, and that d , d 2 ar e close to 1 . Under those assumptions, by Lemma 8.2 we have ( Ω K d ) r 0 ( d 2 − d ) ⊂ ( Ω K d ) α r 0 ( d , d 2 ) ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 11 Setting ǫ = 1 − d and ǫ 2 = 1 − d 2 , we also have (for any k > 0), V k ( r 0 ( ǫ 2 − ǫ )) ∼ κ k − 1 r k − 1 0 ( ǫ 2 − ǫ ) k − 1 . Thus, if Ω K d intersects every plane, by Eq. (6), ( n − 1)2 n − 1 2 V ( C ) ǫ n − 1 2 2 /ω 2 n ≥ κ k − 1 r k − 1 0 ( ǫ 2 − ǫ ) k − 1 /ω k for every ǫ 2 > ǫ. W riting ǫ 2 = ǫ (1 + x ) , we get ( n − 1)2 n − 1 2 V ( C )(1 + x ) n − 1 2 ǫ n − 1 2 /ω 2 n ≥ κ k − 1 r k − 1 0 x k − 1 ǫ k − 1 /ω k , or (7) ǫ n − 1 2 − ( k − 1) ≥ κ k − 1 r k − 1 0 ω 2 n ( n − 1)2 κ k − 1 n − 1 2 V ( C ) x k − 1 (1 + x ) n − 1 2 , for all x > 0 . Applying Lemma 8.1 to the estimate (7), with m = n − 1 2 , l = k − 1 , we see that in order for Ω k 1 − ǫ to intersect every plane thr ough t he origin, we must have (8) ǫ ≥ n + 1 − 2 k 2 κ k − 1 r k − 1 0 ω 2 n ω k V ( C ) ( k − 1) k − 1 ( n − 1) ( n + 1) / 2 ! 2 / ( n + 1 − 2 k ) . If we assume in addition that k ≪ n , the estimate (8) simplifies further to: (9) ǫ > 1 2 r k − 1 0 ω 2 n V ( C ) ! 2 / ( n + 1 − 2 k ) , which simplifies further using Stirling’s formula to: (10) ǫ > 2 π 2 e 2 n 2 r k − 1 0 V ( C ) ! 2 / ( n + 1 − 2 k ) . 6.1. Hyperbolic Spa ce. The corr esponding hyperbolic radius is given by r = 1 2 log  2 − ǫ ǫ  ∼ 1 2 (log 2 − log ǫ ) ∼ 1 2 log 2 − 1 2 log ǫ, so we have Theorem 6.3. Let C ∈ H n be a convex se t of large volume V ( C ) whi c h contains a ball radius r 0 ≪ 1 ar ound a point p ∈ C . T h en if k ≤ ( n − 1) / 2 ther e 12 IGOR RIVIN exists a k-dimensional plane Π k throu gh p , s uch that C ∩ Π k is contained in B ( p , r ) , as long as (11) r = log 2 − 1 2 log( n + 1 − 2 k ) − 1 n + 1 − 2 k ×  ( k − 1) log( k − 1) r 0 + log( κ k − 1 ω 2 n /ω k ) − log V ( C ) − n + 1 2 log( n − 1)  . For n ≫ k , there is the asy m ptotic version: r = log V ( C ) − ( k − 1) log r 0 n + 1 − 2 k + log n − log π − 1 . Remark 6.4 . A comparison of Theorem 6.3 and the estimate (3) in- dicates that we “lose” r oughly 1 2 log n for the diameter of sections of arbitrary convex bodies of volume V versus a ball of the same volume. Example 6.5 . A non-asymptot ic example is when n = 3 , k = 1 . Then we get the estimate (12) r = log 2 − 1 2 log 2 − 1 2  2 log 4 π − log 2 − log V ( C ) − 2 log 2  = 1 2 log V ( C ) − log 2 π, valid for larg e V ( C ) . 7. C onvex sets in E n . Here, we use the techniques developed above to analyze what we can show about convex sets in E n . Related wor k can be found in [4] and r e fe rences ther ein; Klartag’s results ar e asymptotically sharper , but since our methods seem completely di ff erent and more geometric, and the estimates we obtain are quite concret e , the current section seems to be of inter est. Let C be such a convex set, and, as before, we assume that C has positive volume (hence nonempty interior). For simplicity , set p = 0 . Assume that B ( 0 , r 0 ) ⊂ C . It is clear that the diameter of C is bounded, since the volume of a right cone in E n gr ows linearly with the altitude, so the questions about the dimension o f C ∞ do not co me up. However , t he questions of diameter of planar sections as in Section 6 ar e interest ing (especially as they ar e connected to the extensive work on the Busemann-Petty pr oblem, as in [5, 10, 2 , 3 ], and it is not di ffi cult to extend our methods to this setting. ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 13 7.1. V olume estimates. By the standard formulae for Euclide an spher es and balls in Eq. (1) t ogether with C orollary 2.3 we get the follow- ing estimate on the the visual measure of the set Ω r ( C ) of directions wher e the ray of radius r fr om the origin is contained in C : (13) µ ( Ω r ( C )) ≤ V ( C ) κ n r n , wher e , as befor e, V ( C ) d enotes the volume of C . 7.2. Double cone lemma estimates. The pr oof and statement of the Double Cone Lemma 4. 1 go through without change; we st ate the r esult for convenience here: Lemma 7.1. Let r 1 > r 2 > r 0 0 , and let α r 0 ( r 1 , r 2 ) be as in Definition 0 .1. Then the α r 0 ( r 1 , r 2 ) neighborhood on Ω r 1 ( C ) is contained in Ω r 2 ( C ) . 7.3. Applications to finding round sections. As befor e, our basic tool is: Theorem 7.2. There is a k-plane Π k throu gh the origin such that ( Π k ∩ C ) ⊆ B (0 , r ) if µ (( Ω r ) ǫ ) V k − 1 ( ǫ ) < 1 for some ǫ > 0 . Above, V k − 1 ( ǫ ) d e notes the normalized volume of the spherical ball of radius ǫ. Using Eq. (13), Lemma 7.1, and Theorem 7.2, we get: Corollary 7.3. There is a k-plane Π k throu gh the origin, suc h that ( Π k ∩ C ) ⊆ B (0 , r ) , if V ( C ) ω n κ n r n 1 V k − 1 ( α r 0 ( r , r 1 )) < 1 for some 0 < r 1 < r . Combining Lemmas 8.5,8.3,8.2, and 8.1 we see: Lemma 7.4. Assumi ng, as before, that r 1 = r / (1 + x ) , for some x > 0 , (14) min x > 0 V ( C ) ω n κ n r n 1 V k − 1 ( α r 0 ( r , r 1 )) < min x > 0 V ( C )(1 + x ) n ω k κ k − 1 ω n κ n r n r k − 1 0 x k − 1 = V ( C ) ω k κ k − 1 ω n κ n r n r k − 1 0 n n ( k − 1) k − 1 ( n − k + 1) n − k + 1 . 14 IGOR RIVIN And so finally , using Corollary 7.3, we get: Theorem 7.5. Fo r any convex set C ⊂ E n of volume V ( C ) and c ontaining a ball of radius r 0 centered at the origin, an d k ≤ n there is a plane Π k throu gh the origin such that Π k ∩ C ⊆ B n (0 , r ) , for r = n V ( C ) ω k κ k − 1 ω n κ n r k − 1 0 ( k − 1) k − 1 ( n − k + 1) n − k + 1 ! 1 n . Corollary 7.6. If k ≪ n , we can simplify the estimate of Theorem 7.5 to r ∼ n 2 π e V ( C ) 1 / n r ( k − 1) / n 0 .. Note that the radius of the ball in E n of volume V ( C ) is (for large n ) appro ximately r n 2 π e V ( C ) 1 / n , so we lose a factor of √ n / 2 π e . 8. U seful estima tes T o continue, we will need the following: Lemma 8.1. For any x ≥ 0 , and any m > l > 0 , g ( x ) = x l / (1 + x ) m ≤ l l ( m − l ) m − l m m . Proo f . Since g (0) = g ( ∞ ) = 0 , the (smooth) function g ( x ) achieves its maximum at some x 0 in (0 , ∞ ) . Since g ( x ) is positive on the posi- tive r eal axis, its natural logarithm h ( x ) is everywher e defined, and achieves its maximum at x 0 also, since h ′ ( x ) = l / x − m / (1 + x ) , we must have 0 = h ′ ( x 0 ) = l x 0 − m 1 + x 0 = l − ( m − l ) x 0 x 0 (1 + x 0 ) , and so x 0 = l m − l . and g ( x 0 ) =  l m − l  l  1 + l m − l  m . Since 1 + l / ( m − l ) = m / ( m − l ) , the result follows.  ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 15 T o get concr ete estimates, let us write r 1 = r / (1 + x ) , and observe: Lemma 8.2. For x > r 0 , α r 0 ( r , r / (1 + x )) > asin( r 0 x ) . Proo f . It is enough to show that for x ∈ ( r 0 , 1) (15) q r 2 − r 2 0 − q r 2 1 + x 2 − r 2 0 r 1 + x > x , by monotonic ity of asin . The left hand side of Eq. 15 can be rewritten as (1 + x )          s 1 − r 2 0 r 2 − s 1 (1 + x ) 2 − r 2 0 r 2          . By Lemma 8.3, the expr ession inside the parent he se s is smalle r than x / (1 + x ) , a nd so the a ssertion of the Lemma follows.  Lemma 8.3. For a ∈ (0 , 1) , x ∈ ( a , 1) , √ 1 − a 2 − √ x 2 − a 2 > 1 − x , with equality if and only if x = 1 . Proo f . For x = 1 the two sides of the inequality ar e equal to 0 . Other - wise, d ( √ 1 − a 2 − √ x 2 − a 2 ) dx = − x √ x 2 − a 2 = − 1 √ 1 − a 2 / x 2 < − 1 = d (1 − x ) dx , whence the r e sult follows.  Remark 8.4 . The pr oof above actually shows that √ 1 − a 2 − 1 − a > √ 1 − a 2 − √ x 2 − a 2 − 1 − x > 0 for the intervals in question (since the derivat ive of the middle ex- pr e ssion is strictly negative, and the left and right expressions are the values at the two end p oints of the interval ( a , 1) . ) Lemma 8.5. Let V l ( r ) be the normalized volume of the spherical ball of radius r . Then, V l ( r ) > κ l ω l + 1 sin n r Proo f . W e know that V l ( r ) = ω k ω k + 1 Z r 0 sin l − 1 η d η. 16 IGOR RIVIN Making the substitutio n η = asin ρ, we see that V k ( r ) = ω k ω k + 1 Z sin r 0 ρ n − 1 p 1 − ρ 2 d ρ > ω k k ω k + 1 sin k r = κ k ω k + 1 sin k r .  9. E xplicit lower bound on limitset dimension in H 3 . In this section we pr ove Theorem 9.1. F or any β < 1 , there is a set S β in ∂ H 3 , such that Hausdor ff dimension of S β equals β, whi le the volume of the convex hull of S β is finite. Theor em 9 . 1 shows that our dimension estimate is sharp, although it does leave open: Question 9.2 . Does ther e e xist a se t S 1 ⊂ ∂ H 3 with Hausdor ff dimen- sion of S 1 equal to 1 and such that the convex hull of S 1 has finite volume? The proof of Theorem 9.1 is by explicit constructio n, and is con- tained in section 9.2. The ne eded fact s concerning the vo lume of ideal simplices in H 3 ar e contained in section 9.1. 9.1. Ideal simplices in H 3 . Consider B 3 (0 , 1) , viewed as the Klein model of H 3 , and consider points (0 , 0 , 1) , (0 , 0 , − 1 ) , (1 , 0 , 0) , (cos θ, sin θ, 0) on the sphe r e ∂ B 3 (0 , 1) . The convex hull of these four points is an ideal tetrahedron T θ . Under ster eographic pr ojection (fro m the north pole (0 , 0 , 1) the four points go to ∞ , 0 , 2 , 2 exp i θ, and so the dihedral angles of T θ ar e θ, π/ 2 − θ/ 2 , π/ 2 − θ/ 2 . It follows that the hyperbolic volume of T θ is given by (16) V ( T θ ) = L ( θ ) + 2 L ( π/ 2 − θ/ 2) , wher e L ( x ) denotes the Lobachevsky function: (17) L ( x ) = − Z x 0 log 2 | sin t | dt . Many properties and applications of the volumes of ideal simplices ar e discussed in [9], but her e we will only need the following simple r esult: Lemma 9.3. When θ ≪ 1 , we have V ( T θ ) ∼ − θ log θ. ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 17 Proo f . It is clear (for geometric reasons ) that lim θ → 0 V ( T θ ) = 0 . Using Eq. (16), we see that this implies that L ( π/ 2) = 0 , and the statement of the Lemma then follows fr om the fundamental theorem of calculus.  9.2. V olume of the convex hull of Cantor sets. W e will be using the “standard” family of C antor sets C ( α ) , wher e 0 < α < 1 / 2 . Such a set is obtained by starting with the interval [0 , 1] , then deleting the interval ( α, 1 − α ) , then applying the constr uction to e ach of the r ema ining intervals, and so on r e cursively . The usual middle thirds Cantor set is C (1 / 3) . I t is we ll known that the Hausdor ff dimension of C ( α ) equals log 2 log 1 α , see [6] for proo f and discussio n. Consider the set S α ⊂ ∂ B 3 (0 , 1) consisting of the North p ole – (0 , 0 , 1), the South pole – (0 , 0 , − 1) , and a Cantor set on the equator . An example of such a Cantor set can be obtained by identifying the set ( x , y , 0) , y ≥ 0 ⊂ ∂ B 3 (0 , 1) with the unit interval, and then constructing a Cantor set C ( α ) in that interval.. W e claim that the convex hull of S ha s finite volume. Indeed, the convex hull of S is a union of ideal tetrahedra T θ , as above. Each tetrahedron corr esponds to an interst itial r egion in the Cantor c onstruction, so that, for example, the first stage cont ributes a T π (1 − 2 α ) , the second stage contributes two copies of T πα (1 − 2 α ) , and the n -th stage contributes 2 n − 1 copies of T π/ (1 − 2 α ) α n − 1 . It follows that the volume of the convex hull of S is: (18) ∞ X n = 1 2 n − 1 V ( T π (1 − 2 α ) α n − 1 ) . For su ffi ciently large n , Lemma 9.3 tells us that V ( T π/ (1 − 2 α ) α n − 1 ) is of the or der of − ( n − 1) α log α, a nd so the sum in Eq. (18 ) converges, thus the volume is finite. Note, however , that the volume of the convex hull of S α goes to infinity as α tends to 1 / 2 , a nd thus the Hausdor ff dimension of C ( α ) tends to 1 . 10. H igher D imension The lower bounds for n = 3 see m to depend on an explicit for- mula for the vo lume of ide al simplices and the geometry of one- dimensional Cantor sets. I t turns out that both aspects can be gener- alized to higher dimensions. The sets we will use will be generali zed Sierpinski carpets , K ( M ) , constr ucted as follows: 18 IGOR RIVIN W e start with the unit cube K 0 = K n = [0 , 1] n ⊂ R n . At the next step we subdivide K 0 into N n equally sized cubes, each of side-length 1 / N . Number t he se cubes fr om 1 to N n . If M ⊆ 1 , . . . , N n , de lete all the cubes whose indices are not in M , to obtain the set K 1 ( M ) . Now , apply the p rocess to each of the M remaining cubes to obtain K 2 ( M ) , and so on. The final carpet is the limiting object: K ( M ) = ∞ \ k = 0 K k ( M ) . The standard Sierpinski carpet is obtained by setting n = 2 , N = 3 , M = { (1 , 1) , (1 , 2) , (1 , 3 ) , (2 , 1 ) , (2 , 3) , (3 , 1 ) , (3 , 2) , (3 , 3) } , where the num- bering goes fr om top right to bottom left. The unit interval can be obtained in this setting by letting M = { (2 , 1) , (2 , 2) , (2 , 3) } . The middle thir ds Cantor set is obtained by setting n = 1 , N = 3 , M = { 1 , 3 } , the numbering going fr om left to right. The following Theor e m follows immediately from [6, Section 4.12]: Theorem 10.1. The Hausdor ff dimension of K ( M ) equals log | M | / log N . Theor em 10. 1 gives us the we ll-known values log 2 / log 3 , = 0 . 63 , log 8 / log 3 = 1 . 8 9 , 1 for the H a usdor ff dimensions of the Sierpinski carpet, the middle thir ds Cantor set, and the unit interval, r espec- tively . Theor em 10.1 has the following obvious cor ollary: Corollary 10.2. In R n there are Sierpinski carpets of H ausdor ff dimension arbitrarily close (but not equal) to n . Proo f . Let N = 2 L , and let M be the set of n -tuples ( i 1 , . . . , i n ) wher e i k = k mod 2 . The cardinality of M e q uals L n , a nd so the Hausdor ff dimension of K ( M ) equa ls n − n log 2 / log N . For N ≫ n , this will be close to n .  The set in the example above is constr ucted in such a way that the sets at the k -iteratio n of the construct ion have diameter exponentially decr e asing wit h k . The same constr uction as in thr ee dimensions ( mutatis mutandis ) gives us the result that in any odd dime nsion, our bound on the Hausdor ff dimension is sharp, in other wor d s: Theorem 10.3. For any β < 1 , the re is a set S β in ∂ H 3 , s uch that H ausdor ff dimension of S β equals β, whi le the volume of the convex hull of S β is finite. R eferences [1] Bruno Colbois a nd Patrick V erovic. Hilbert geometry for strictly convex do- mains. Geom. Dedicata , 105:2 9–42, 2004. ASYMPTOTICS OF CONVE X SETS IN E n AND H n . 19 [2] R. J. Gardner . A positive answer to the Busemann-Petty problem in thr ee dimensions. Ann. of Math. (2) , 1 40(2) :435– 4 47, 1994. [3] Eric Grinberg a nd Igor Rivin. Infinitesimal aspects of the busemann-petty problem. B ulletin of the London Mathematical Society , 22(5) : 478–4 84, 1990 . [4] Bo’az Klartag. A geometric inequality and a low M -estimate. Proc. Amer . Math. Soc. , 132 (9):26 19–2628 (electronic), 200 4. [5] Alexa nde r Koldobsky . The Busemann-Petty problem via spherical harmonics. Adv . Math. , 177(1) :105–1 14, 2003. [6] Pertti Mattila. Geomet ry of sets and measur es in Euclidean spaces , volume 44 of Cambridge Studies in Ad vanced Mathematics . Cambridge University Press, Cambridge, 1995. Fractals and rectifiability . [7] V itali D. Milman and Gideon Schechtman. Asymp totic theory of finite- dimensional normed sp aces , volume 1200 of Lecture Notes in Math ematics . Springer-V erlag, Berlin, 1986. W ith an appe ndix by M. Gromov . [8] John W . M ilnor . How to compute volume in hyperbolic space , volume 1. Publish or Perish, Houston, T exas, 1994. [9] Igor Rivin. Euclidean structures on simplicial surfaces a nd hyperbolic volume. Annals of Mathematics (ser . 2) , 1 39(3) :553– 5 80, May 1 994. [10] Gaoyong Zhang. A positive solution to the Busemann-Petty problem in R 4 . Ann. of Math. (2) , 149(2) : 535–5 43, 1999. D ep artment of M a th ema tics , T emple U niversity , P hiladelphia E-mail address : rivin @math. temple.edu