Bounds on the roots of the Steiner polynomial

Bounds on the ro ots of the Steiner P olynomial Madeleine Jetter No vem b er 2, 2018 Abstract W e consider the Steiner p olynomial of a C 2 con vex b ody K ⊂ R n . Denote b y ρ min the minim um v alue of the principal radii of curv ature of ∂ K and b y ρ max their maxim um. When n ≤ 5, the real parts of the ro ots are b ounded ab o ve by − ρ min and b elow by − ρ max . These b ounds are v alid for an y n such that all of the roots of the Steiner p olynomial of ev ery con vex b ody in R n lie in the left half-plane. 1 In tro duction Let K ⊂ R n b e a (compact) con vex b o dy and let B denote the unit ball in R n . W e form the outer parallel b o dy K + tB by taking the Mink owski sum of K and a ball of radius t > 0, that is: K + tB = { ~ v + t~ u | ~ v ∈ K, ~ u ∈ B } . Thinking of the outer parallel b ody as the result of the unit-sp eed out ward normal flow applied to K at time t makes it relev ant to applied problems such as combustion [ 1 ]. The volume of K + tB can b e written as a p olynomial of degree n , the Steiner p olynomial [ 4 ]: V K + tB = n X i =0  n i  V ( K n − i , B i ) t i where the coefficient V ( K n − i , B i ) is the mixed v olume of n − i copies of K and i copies of the unit ball. W e will adopt the notation S K ( t ) = V K + tB for the Steiner p olynomial of K in the v ariable t . In t wo dimensions, consideration of the roots of the Steiner polynomial leads to a Bonnesen-style inequality . When K is a conv ex planar region, with area A K and p erimeter L K , we hav e S K ( t ) = A K + L K t + π t 2 . Since the discriminant of the Steiner polynomial in tw o dimensions is L 2 K − 4 π A K , we see that the isop erimetric inequality for K is equiv alent to the fact 1 that S K ( t ) = 0 has (one double or t wo single) real ro ots. Moreov er, since S K giv es the area of the region K + tB , the ro ots must also b e negativ e when A K > 0. F urthermore, it is known that Theorem 1.1. L et K b e a strictly c onvex r e gion which is not a disc. L et R i = sup { r | a tr anslate of r B ⊂ K } b e the inr adius of K , and let R e = inf { r | a tr anslate of K ⊂ r B } b e the outr adius. L et ρ min and ρ max denote the minimum and maximum values of the r adius of curvatur e of K . If the r o ots of S K ar e t 1 < t 2 , then − ρ max < t 1 < − R e < − L K 2 π < − R i < t 2 < − ρ min . (1) When K is a disc, then all of the ab ov e quan tities are equal, giving a version of Bonnesen’s inequality . Green and Osher pro vide a pro of in [ 1 ]. T eissier [ 5 ], working in the setting of ample divisors on algebraic v arieties, p osed the following problems aimed at generalizing th e app ealing state of affairs in the planar case. Supp ose a conv ex b ody K ⊂ R n is giv en and that the ro ots of S K ha ve real parts r 1 ≤ r 2 ≤ · · · ≤ r n . P1. Is S K stable (i.e. do all the ro ots lie in the left half-plane)? P2. Let R i indicate the inradius of K , that is, the largest real num b er s suc h that a translate of sB is contained in K . Do es the inequalit y − R i ≤ r n hold? By the Routh-Hurwitz stability criterion and the Aleksandrov-F enchel in- equalities [ 4 ], w e kno w that S K is stable for all con vex b odies K ⊂ R n pro vided that n ≤ 5. On the other hand, Cifre and Henk construct an example in [ 2 ] to show that S K need not be stable when K ⊂ R 15 . Less is known ab out the inradius b ound. Ho w ever, in those cases where T eissier’s first problem has an affirmativ e answer, w e can prov e a generalization of the extreme upp er and lo wer bounds in inequalit y ( 1 ) relatively easily . Theorem 1.2. Assume that in R n , S K is stable for every c onvex K ⊂ R n . L et K ⊂ R n b e a C 2 c onvex b o dy, and supp ose that the r o ots of S K have r e al p arts r 1 ≤ r 2 ≤ · · · ≤ r n . Denote by ρ min and ρ max the minimum and maximum values of the princip al r adii of curvatur e of K . Then − ρ max ≤ r 1 ≤ · · · ≤ r n ≤ − ρ min . 2 T ec hnical Background 2.1 The Steiner P olynomial A general reference for this section is Schneider’s volume [ 4 ]. The fundamental to ol for what follo ws is the support function p K : R n → R of a conv ex bo dy K ⊂ R n , defined as follows: 2 p K ( ~ x ) = sup { ~ x · ~ v | ~ v ∈ K } , where · denotes the standard inner product. Because of the homogeneit y of the supp ort function, p K is determined by its restriction to the unit sphere. Th us we frequen tly treat p K as a function on S n − 1 . A particularly imp ortan t feature of p K is the wa y in which it carries informa- tion ab out the curv ature of ∂ K when the b oundary satisfies certain smo othness conditions. When K (and thus p K ) is C 2 , w e consider the Hessian matrix H ( p K ). Given ω ∈ S n − 1 , we choose a basis { e 1 , . . . , e n } where { e 1 , . . . , e n − 1 } is an orthonormal basis for T S n − 1 ω and e n = ω . One can sho w using homogeneit y [ 4 ] that the eigenv alues of H ( p K ( ω )) computed with resp ect to this basis are 0 and the principal radii of curv ature of K at ω , which we denote ρ 1 , . . . , ρ n − 1 . The restriction of the Hessian to T S n − 1 ω , which w e write as H ( p K ( ω )), has eigen v alues ρ 1 , . . . , ρ n − 1 . Since the infinitesimal element of area on ∂ K is the pro duct of the principal radii of curv ature, w e may write the volume of K equiv alently as V K = 1 n Z S n − 1 p K ρ 1 · · · · · ρ n − 1 dω or V K = 1 n Z S n − 1 p K det H ( p K ) dω where H denotes the Hessian matrix computed with respect to an orthonormal frame for T S n − 1 . Applying this formula to K + tB , noting that p K + tB = p K + t , we hav e S K = V K + tB = 1 n Z S n − 1 ( p K + t ) det  H ( p K ) + tI  dω . (2) The integrand ab o v e is a p olynomial of degree n in t . W e can isolate the co efficien t of each t i using the Minko wski in tegral formulas ([ 4 ], p. 291) to obtain an integral expression for V ( K n − i , B i ). V ( K n − i , B i ) = 1 n Z S n − 1 s n − i ( ρ 1 , . . . , ρ n − 1 ) dω = 1 n Z S n − 1 p K s n − i − 1 ( ρ 1 , . . . , ρ n − 1 ) dω , where s j is the normalized j th elemen tary symmetric function in ρ 1 , . . . , ρ n − 1 (ie.  n − 1 j  s j is the usual j th elemen tary symmetric function). 2.2 Mink o wski Subtraction The proof of theorem 1.2 will also rely on the concept of Minko wski subtraction. Giv en conv ex b o dies K , L ⊂ R n , the Minkowski differ enc e of K and L is K ∼ L = { ~ v ∈ R n | L + ~ v ⊂ K } . 3 W e ma y think of K ∼ L as the intersection of all translates of K b y opposites of vectors in L . If K and L are b oth con vex, then K ∼ L is as well, but the operations of Mink owski sum and difference are not inv erse to one another. Although ( K + L ) ∼ L = K holds for an y conv ex b o dies K and L , ( K ∼ L ) + L = K only when there exists a con vex b o dy M such that L + M = K . In this case w e say that L is a Minko wski summand of K , and ( K ∼ L ) + L = M + L = K . Sp ecializing to a situation relev ant to the pro of, when w e know that cB is a Minko wski summand of K , we hav e that ( K ∼ cB ) + cB = K and it follo ws that p K ∼ cB = p K − c . This allo ws us to compute S K ∼ cB fairly easily using Equation ( 2 ). W e will make use of the follo wing lemma appearing in [ 3 ] and [ 4 ], whic h giv es a condition under whic h L is a Minko wski summand of K . Lemma 2.1. Supp ose K , L ⊂ R n ar e c onvex. If the maximum of al l the prin- cip al r adii of curvatur e of L is b ounde d ab ove by the minimum of the princip al r adii of curvatur e of K at e ach ω ∈ S n − 1 , then L is a Minkowski summand of K – i.e. ther e is a c onvex b o dy M such that L + M = K . 3 Pro of of Theorem 1.2 W e first establish the upp er bound, which is the easier of the tw o. Since K is conv ex, eac h ρ i ≥ 0. W e may assume that K is C 2 + , (in other words the principal radii of curv ature are all strictly positive and hence ρ min > 0) since otherwise there is nothing to pro ve. If 0 ≤ c ≤ ρ min , then let K 0 = K ∼ cB . cB is a Minko wksi summand of K by Lemma 2.1 , so S K 0 ( t ) = 1 n Z S n − 1 ( p K − c + t )det  H ( p K ) + ( − c + t ) I  dω = S K ( t − c ) . The ro ots of S K 0 ha ve real parts r i + c , so the stability assumption implies r i + c < 0, hence r i < − c for an y c ≤ ρ min . Letting c = ρ min yields the claimed upp er b ound. T urning to the low er b ound, let c ≥ ρ max . Then K is a Minko wski summand of cB and we write K 0 = cB ∼ K . W riting p K for the supp ort function of K , w e hav e p K 0 = c − p K . Expanding the Steiner p olynomial of K 0 , 1 n Z S n − 1 ( − p K + c + t )det  H ( − p K ) + ( c + t ) I  dω , in the case n = 3 we hav e S K 0 = −  V K − A K ( c + t ) + H K ( c + t ) 2 − V B ( c + t ) 3  , and in general S K 0 = ( − 1) n S K ( − t − c ). The roots of S K 0 ha ve real parts − ( r + c ), so b y stabilit y − r − c < 0 and w e conclude that − c < r . The lo wer b ound follo ws b y taking c = ρ max . Corollary 3.1. The r e al p arts of the r o ots of S K ar e b ounde d by − ρ min and − ρ max for any C 2 c onvex b o dy K ⊂ R n wher e n ≤ 5 . 4 Pr o of. It is known [ 5 ] that for n ≤ 5, S K is stable for ev ery conv ex b o dy K ⊂ R n . This follows from the Routh-Hurwitz stability criterion and the Aleksandrov- F enchel inequalities. References [1] Mark Green and Stanley Osher. Steiner p olynomials, Wulff flows, and some new isop erimetric inequalities for conv ex plane curves. Asian J. Math. , 3(3):659–676, 1999. [2] Mar ´ ıa A. Hern´ andez Cifre and Martin Henk. Notes on the ro ots of steiner p olynomials. 2007, arXiv:math/0703373v1 . [3] G. Matheron. La form ule de Steiner p our les ´ erosions. J. Appl. Pr ob ability , 15(1):126–135, 1978. [4] Rolf Schneider. Convex b o dies: the Brunn-Minkowski the ory , volume 44 of Encyclop e dia of Mathematics and its Applic ations . Cam bridge Universit y Press, Cambridge, 1993. [5] B. T eissier. Bonnesen-type inequalities in algebraic geometry . I. In tro duction to the problem. In Seminar on Differ ential Ge ometry , volume 102 of Ann. of Math. Stud. , pages 85–105. Princeton Univ. Press, Princeton, N.J., 1982. 5