ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE Abstract. We nvestgate the roots of relatve Stener polynomals of convex bodes. In dmenson 3 we gve a precse descrpton of ther locaton n the complex plane and we study the analogous problem n hgher dmensons. In partcular, we show that the roots (n the upper half plane) form a convex cone; for dmensons 9 ths cone s completely contaned n the (open) left half plane, whch s not true n dmensons 12. Moreover, we characterze certan specal famles of convex bodes by means of propertes of ther roots. 1. Introducton Let K n be the set of all convex bodes,.e., compact convex sets, n the n- dmensonal Eucldean space R n, and let B n be the n-dmensonal unt ball. The subset of K n consstng of all convex bodes wth non-empty nteror s denoted by K0 n. The volume of a set M Rn,.e., ts n-dmensonal Lebesgue measure, s denoted by vol(m). For two convex bodes K, E K n and a non-negatve real number λ, the volume of the so called relatve outer parallel body of K wth respect to E, K + λ E, s expressed as a polynomal of degree n n λ and t can be wrtten as (1.1) vol(k + λe) = W (K; E)λ. =0 Ths expresson s called Mnkowsk-Stener formula or relatve Stener formula of K. The coeffcents W (K; E) are the relatve quermassntegrals of K, and they are a specal case of the more general defned mxed volumes for whch we refer to [19, s. 5.1]. In partcular, we have W 0 (K; E) = vol(k), W n (K; E) = vol(e) and W (K; E) = W n (E; K). If E = B n, (1.1) becomes the classcal Stener formula [20], and W (K; B n ), for short denoted by W (K), s the classcal -th quermassntegral of K. 2000 Mathematcs Subject Classfcaton. Prmary 52A20, 52A39; Secondary 30C15. Key words and phrases. Roots of Stener polynomal, tangental bodes, constant wdth sets, sausages. Second author s supported n part by Subdreccón General de Proyectos de Investgacón (MCI) MTM2009-10418 and by Programa de Ayudas a Grupos de Excelenca de la Regón de Murca, Fundacón Séneca, 04540/GERM/06. 1

2 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE In the followng, E K0 n wll be always a fxed convex body wth nonempty nteror, and for K K n we wll wrte f K;E (z) = W (K; E)z =0 to denote the Stener polynomal of K relatve to E, regarded as a formal polynomal n a complex varable z C. We are nterested n the locaton of the roots of f K;E (z). To ths end let C + = {z C : Im(z) 0} be the set of complex numbers wth non-negatve magnary part, and let (1.2) R(n, E) = { z C + : f K;E (z) = 0 for some K K n} be the set of all roots of f K;E (z), K K n, n the upper half plane. Theorem 1.1. R(n, E) s a convex cone contanng the non-postve real axs. By the sopermetrc nequalty (cf. e.g., [19, p. 318]) t s easy to see that n the plane, R(2, E) s exactly the non-postve real axs. In order to descrbe the 3-dmensonal cone we need the so called cap-bodes. A convex body K K n s called a cap-body of L K n f K s the convex hull of L and countably many ponts such that the lne segment jonng any par of these ponts ntersects L. Wth ths notaton we can precsely descrbe R(3, E) (see Fgure 1). Theorem 1.2. If E K0 3 s a cap-body of a planar convex body then { R(3, E) = x + y C + : x + } 3 y 0, otherwse R(3, E) = { x + y C + : x + } 3 y < 0 {0}. 2 1-4 -3-2 -1 Fgure 1. Roots of 3-dmensonal Stener polynomals n the upper half plane. In partcular, we note that the set R(2, E), as well as the closure of R(3, E), are ndependent of the gauge body E. It seems to be qute lkely

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 3 that ths holds true n any dmenson. Our result s restrcted to dmenson 3 snce n hgher dmenson we do not have enough nformaton about the so called complete system of nequaltes among the quermassntegrals (cf. e.g., [6, Problem 6.1]). We also do not know whether n general R(n, E) R(n+1, E) or whether for partcular gauge bodes (e.g. E = B n ) the cone s always half open as n the case n = 3. The property that all the roots of 3-dmensonal Stener polynomals le n the left half plane was part of a conjecture posed by Sangwne-Yager [17] (cf. e.g., [18, p. 65]) whch was motvated by a problem of Tesser [21]. There t was n partcular clamed that Stener polynomals are weakly stable polynomals,.e., R(n, E) { z C + : Re(z) 0 }. Ths ncluson s known to be true for dmensons 5 (cf. e.g., [21]), but n [7] t was shown to be false n dmensons 12 for a specal famly of 3- tangental bodes (see also [13] for another famly of hgh dmensonal convex bodes wth ths property). Here we narrow the gap between these values of the dmenson by showng the followng result. Proposton 1.1. ) For n 9 we have R(n, E) { z C + : Re(z) < 0 } {0}. ) For n 12 we have { z C + : Re(z) 0} R(n, E). For further nformaton on the roots of Stener polynomals n the context of Tesser s problem we refer to [7, 8, 10, 11, 13]. Next we consder the problem to characterze convex bodes by propertes of the roots of ther Stener polynomals. To ths end we denote by r(k; E) = max{r 0 : x R n wth x + r E K} the nradus of K K n wth respect to E. If E = B n then r(k; B n ), for short r(k), s the classcal nradus,.e., the radus of a largest Eucldean ball contaned n K. It s easy to check that for an l-dmensonal convex body L K n, the Stener polynomal of K = L + r B n has the (n l)-fold root r = r(k). We beleve that those bodes are completely characterzed by ths property. Conjecture 1.1. Let K K n and let l {0,..., n 1}. Then r(k) s an (n l)-fold root of f K;Bn (z) f and only f K = L + r(k)b n for some L K n wth dm L = l. The case l = 0 s known to be true (see [9]) and wll also follow from a more general statement n Secton 3 (Proposton 3.2). The case l = n 1 s related to a conjecture of Matheron [15] (see also [19, p. 212]) clamng that B n s a summand of K K n,.e., K = L + B n for some L K n, f and only f the volume of the nner parallel body K λb n = {x K : x + λ B n K} s gven by the so called alternatng Stener polynomal,.e., (1.3) vol(k λb n ) = f K;Bn ( λ), λ (0, 1).

4 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE In fact, ths was conjectured by Matheron for any gauge body E K n 0, not only for B n. Observe that for K K n wth r(k) = 1, (1.3) mples that r(k) s a root of f K;Bn (z). So far Matheron s conjecture s only known to be true n the case n = 2 (see [15]). Based on the equalty case n Bonnesen s nequalty, namely, (1.4) W 0 (K; E) 2W 1 (K; E)r(K; E) + W 2 (K; E)r(K; E) 2 0 wth equalty f and only f K = L + r(k; E)E for dm L 1 (see [1, pp. 33 36], [3]), we also see that Conjecture 1.1 s true for any gauge body E n dmenson 2. Regardng the sausage case l = 1 of the conjecture above we are able to prove the followng weakenng. Theorem 1.3. Let K K n, n 2. Then r(k) s an (n 1)-fold root of f K;Bn (z) and all ts 2-dmensonal projectons onto any 2-dmensonal lnear subspace have nradus r(k) f and only K s a sausage wth nradus r(k),.e., there exsts a lne segment L such that K = L + r(k)b n. We note that, for nstance, cap-bodes of B n (see Secton 3 for the defnton) have the property that all ther 2-dmensonal projectons have the same nradus as the body, namely, 1 = r(b n ). In the last secton we show that cap-bodes, as well as the more general class of p-tangental bodes, can be characterzed va the roots of ther Stener polynomals (see Proposton 3.1 and Corollary 3.1). Moreover, we nvestgate the roots of Stener polynomals of constant wdth sets and obtan the followng result. Proposton 1.2. The roots of Stener polynomals f K;Bn (z) of convex bodes K of constant wdth b are symmetrc wth respect to b/2,.e., f K;Bn (z) = 0 f and only f f K;Bn ( z b) = 0. The paper s organzed as follows. At the begnnng of Secton 2 we gve some prelmnary results on propertes of roots of Stener polynomals, whch are then needed for the proofs of Theorems 1.1 and 1.2 and Proposton 1.1. In Secton 3 we provde the announced characterzaton of cap-bodes and p- tangental bodes and we present the proofs of Theorem 1.3 and Proposton 1.2; both results wll be consequences of more general statements. 2. On the locaton of the roots of Stener polynomals Frst we collect some propertes on the behavor of the roots of Stener polynomals when we change a bt the nvolved bodes. Lemma 2.1. Let γ be a root of the Stener polynomal f K;E (z). ) Let λ > 0. Then λ γ s a root of f λ K;E (z). ) Let a 0. Then γ a s a root of f K+a E;E (z). ) Let γ = x + y wth x < 0, and let 0 < ρ 1. Then x + (ρ y) s a root of f ρ K+x(ρ 1) E;E (z).

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 5 Proof. By the homogenety of the quermassntegrals we have W (λ K; E) = λ n W (K; E) for = 0,..., n, and hence f λk;e (z) = λ n f K;E (z/λ), whch shows ). Snce for any a, λ 0 vol(k + a E + λ E) = vol ( K + (a + λ) E ) = W (K; E) (a + λ), we have f K+aE;E (z) = f K;E (a + z), whch mples ). Fnally, ) s just a combnaton of ) and ), snce ρ K +x(ρ 1) E = ρ ( K +[x(ρ 1)/ρ] E ). Next we gve the proof of Theorem 1.1. Proof of Theorem 1.1. Frst we note that 1 s a root of the polynomal f E;E (z) snce n ths case W (E; E) = W 0 (E; E) for all = 1,..., n, and thus f E;E (z) = W 0 (E; E)(z + 1) n. Hence, by Lemma 2.1 ) the negatve real axs s contaned n R(n, E). Next observe that also 0 s a root of f K;E (z) for any K K n wth dm K < n, because n ths case W 0 (K; E) = 0. Together wth Lemma 2.1 ) ths shows that R(n, E) s a rayset,.e., for γ R(n, E) and any λ 0 we also have λ γ R(n, E). Hence, wth Lemma 2.1 ) ths shows that R(n, E) s a convex cone. In fact, gven roots γ = x + y R(n, E) of polynomals f K ;E(z), = 1, 2, and ρ (0, 1), we can construct a convex body M such that ργ 1 + (1 ρ)γ 2 s a root of f M;E (z) as follows. Snce R(n, E) contans the non-postve real numbers but no postve numbers, we may assume that not both roots are real numbers. Let x 1 /y 1 = max{x /y : y > 0, = 1, 2} and let µ = y 1 / ( ) ρ y 1 + (1 ρ) y 2. By the choce of x 1 /y 1 we have =0 ν = µ ( ρ x 1 + (1 ρ) x 2 ) = y1 ρ x 1 + (1 ρ) x 2 ρ y 1 + (1 ρ) y 2 x 1, snce t s easy to see that the above functon s ncreasng n ρ (0, 1). So we know by Lemma 2.1 ) that ν + y 1 s a root of f K1 +(x 1 ν)e;e and thus, by Lemma 2.1 ) ργ 1 + (1 ρ)γ 2 s a root of the Stener polynomal of M = (1/µ) ( K 1 + (x 1 ν)e ). In [7, Theorem 1.2, Remark 3.2] t was shown that for dmensons n 12 there exst convex bodes K such that some of the roots of f K;Bn (z) have postve real part. The constructons of these so called 3-tangental bodes of B n can be easly generalzed to an arbtrary gauge body E K n 0. Hence we know by Theorem 1.1 that, for n 12, { z C + : Re(z) 0 } R(n, E), whch s the second statement n Proposton 1.1. The frst statement of ths proposton descrbes a stablty property of Stener polynomals. We recall that a real polynomal s sad to be stable f all ts roots have (strct) negatve real part. Hence, agan on account of Theorem 1.1, Proposton 1.1 ) says that for dmenson n 9 Stener

6 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE polynomals are stable f we neglect for a moment the root 0. The man ngredent n order to prove t are the nequaltes (2.1) W (K; E) 2 W 1 (K; E)W +1 (K; E), 1 n 1, (2.2) W (K; E)W j (K; E) W k (K; E)W l (K; E), 0 k < < j < l n, partcular cases of the Aleksandrov-Fenchel nequalty (see e.g. [19, s. 6.3]). Proof of Proposton 1.1. It remans to show part ). We know 0 s a root of f L;E (z) f and only f dm L = l < n, and n ths case we may wrte f L;E (z) = z n l n ( n =n l ) W (L; E)z n+l wth W (L; E) > 0 for n l n. Hence we have to show that all polynomals of the type n ( n =n l ) W (L; E)z n+l, W (L; E) > 0 for n l n, 0 < l n 9, are stable. For n 5 t was already ponted out by Tesser [21, p. 103] that nequaltes (2.1) and (2.2) guarantee that the Routh-Hurwtz crteron for stablty of polynomals (see e.g. [14, p. 181]) s fulflled, and so t remans to consder the case n 6. Here we use the followng stablty crteron [16, Theorem 3] (see also an ndependent proof n [12, Theorem 1]). A real polynomal f(z) = n =0 a z, wth a > 0 for = 0,..., n, s stable f a 1 a +2 β a a +1, = 1,..., n 2, where β 0.4655 s the only real soluton of z(z + 1) 2 = 1. Agan t s easy to check that (2.1) and (2.2) mply ths crteron. So only n dmensons n = 10, 11 we do not know whether Stener polynomals can have roots wth postve real parts. Obvously, by the convexty of the set R(n, E) the exstence of a root wth postve real part also mples the exstence of a pure magnary complex root. However, not all roots can be of that type. More precsely Proposton 2.1. There exsts no convex body K K n such that all roots of f K;E (z) are magnary pure complex numbers (excludng the real root always exstng n odd dmenson). Proof. Notce frst that f dm K 1 then the Stener polynomal s of degree 1 and there s nothng to prove. Thus we can assume that dm K 2. For n even, let K K n be a convex body such that all roots of f K;E (z) are {±y j, j = 1,..., n/2}, wth y j R >0. Then we get n/2 f K;E (z) = W n (K; E) (z 2 + yj 2 ), whch mples W 2+1 (K; E) = 0 for all = 0,..., (n 2)/2. In partcular, W n 1 (K; E) = 0 and hence dm K = 0, whch contradcts our assumpton. For n odd, let K K n be a convex body such that the roots of f K;E (z) are { x, ±y j, j = 1,..., (n 1)/2}, wth x 0 and y j R >0. From j=1 (n 1)/2 f K;E (z) = W n (K; E)(z x) (z 2 + yj 2 ) j=1

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 7 we get (n 1)/2 j=1 y 2 j = Wn 2 (K; E) 2 W n (K; E), x = n W n 1(K; E) W n (K; E). (n 1)/2 x j=1 y 2 j = Wn 3 (K; E) 3 W n (K; E), Thus we obtan the relaton 3n n 2 W n 2(K; E)W n 1 (K; E) = W n (K; E)W n 3 (K; E). For dm K 2 the left hand sde n the above equalty s postve, and snce t holds W n 2 (K; E)W n 1 (K; E) W n (K; E)W n 3 (K; E) for all convex bodes (cf. (2.2)), we get the desred contradcton. In Secton 3 we wll show a knd of generalzaton of the proposton above, provng that all roots of f K;E (z) cannot le on two symmetrc parallel lnes of the form {x + y : x = ± a}, except for the case of an n-fold real root (Proposton 3.2). Next we come to the proof of Theorem 1.2 n whch we gve a descrpton of the cone R(3, E) (see Fgure 1). Proof of Theorem 1.2. Let a + b C + be a root of a Stener polynomal f K;E (z) for some K K 3 and E K0 3. By Proposton 1.1 we may assume that both a, b > 0 and we have to show that 3 b a. Moreover, let c, c 0, be the real root of f K;E (z). Then we have the denttes (2.3) 2a+c = 3 W 2(K; E) W 3 (K; E), a2 +b 2 +2ac = 3 W 1(K; E) W 3 (K; E), c(a2 +b 2 ) = W 0(K; E) W 3 (K; E). Inequaltes (2.1) for = 2 and = 1 yeld n terms of a, b, c, respectvely, (2.4) (a c) 2 3b 2, (2.5) (a 2 3b 2 )c 2 2a(a 2 + b 2 )c + (a 2 + b 2 ) 2 0. The frst one s equvalent to c a 3 b or c a + 3 b. In the frst case we are done and so we are left wth the case c a + 3 b. Suppose t would be a < 3 b. Then the left hand sde of (2.5) s a quadratc polynomal n c wth negatve leadng coeffcent, and wth zeros (a 2 + b 2 )/(a 3 b) and (a 2 +b 2 )/(a+ 3 b). Thus, on account of c 0, nequalty (2.5) s equvalent to c (a 2 + b 2 )/(a + 3 b). Hence we obtan the contradcton a + 3 b c a2 + b 2 a + 3 b, and therefore, a 3 b. Thus we have shown the ncluson R(3, E) { a + b C + : 3 b a } for any E K n 0.

8 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE Next we study the boundary,.e., we suppose that we have 3 b = a. In ths case, (2.4) and (2.5) reduce to c(c 2a) 0 and c (2/3)a, respectvely, because a > 0. Snce c 0 we must have c = 0 and thus W 0 (K; E) = 0. Then, n vew of W 1 (K; E) > 0 ths shows that dm K = 2. Moreover, by (2.3) we fnd W 2 (K; E) = (2/3) a W 3 (K; E) and W 1 (K; E) = (4/9) a 2 W 3 (K; E), and so W 2 (K; E) 2 W 1 (K; E)W 3 (K; E) = 0. Now t s known (see [2]) that the above equalty holds f and only f E s a cap-body of the 2-dmensonal set K. Thus f E K0 3 s not a cap-body of a planar set we have the ncluson R(3, E) { a + b C + : 3 b < a } {0}. In order to conclude the proof of the theorem we have to show that for any a + b C +, wth a > 3 b or a = 3 b, dependng on the gauge body E K0 n, there exsts K Kn such that f K;E ( a + b) = 0. We have to dstngush three cases. If b = 0 then we just have to consder K = ae for a > 0 and any planar convex body for a = 0. So let b > 0 and a > 3 b. Here we take a sutable dlate of a cap-body of E. Indeed the complex roots of the Stener polynomals of all possble cap-bodes of E (see Remark 3.1 when k, n = 3) determne a contnuous parametrzed curve ( t + 2 2(t 2 + t + 1), ) 3 t 2(t 2, for t (0, 1), + t + 1) wth lmt ponts ( 1, 0), ( 1/2, 3/6) when t tends to 0 and 1, respectvely (see Fgure 2). By Lemma 2.1 ) the complex number a + b wll be the root of the Stener polynomal of some dlate of a cap-body of E. Fnally, let b > 0 and a = 3 b. Here E K0 n has to be a cap-body of a planar convex body K, and then a dlate of K tself wll gve the soluton. In fact, snce E s a cap-body of K then vol(e) = W 0 (E; K) = W 1 (E; K) = W 2 (E; K) ([19, p. 368, proof of Theorem 6.6.16]), and so notcng that W 0 (K; E) = 0 we get f K;E (z) = z 3 =1 ( ) 3 W (K; E)z 1 = z = z vol(e)(3 + 3z + z 2 ), 3 =1 ( ) 3 W 3 (K; E)z 1 wth non-zero roots 3/2 ± 3/2. Lemma 2.1 ) concludes the proof. Just computng the roots of the Stener polynomal of 4-dmensonal capbodes (see agan Remark 3.1) we see that R(3, E) R(4, E) strctly. 3. Characterzng convex bodes by propertes of the roots Ths secton s devoted to characterze (famles of) convex bodes by means of propertes of the roots. Here we wll need the nequaltes (3.1) W (K; E) r(k; E)W +1 (K; E), {0,..., n 1},

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 9 whch are a drect consequence of the monotoncty of the mxed volumes (cf. e.g. [19, p. 277]), snce, up to translatons, r(k; E)E K. I. p-tangental bodes. A convex body K K n contanng E K n 0 s called a p-tangental body of E, p {0,..., n 1}, f each support plane of K that s not a support plane of E contans only (p 1)-sngular ponts of K. Here a boundary pont x of K s sad to be an r-sngular pont of K f the dmenson of the normal cone n x s at least n r. For further characterzatons and propertes of p-tangental bodes we refer to [19, Secton 2.2]. So a 0-tangental body of E s E tself, 1-tangental bodes are just the cap-bodes (see [19, p. 76]) and each p-tangental body of E s also a q- tangental body for p < q n 1. If K s a p-tangental body of E then r(k; E) = 1, and the followng result by Favard states a characterzaton of n-dmensonal p-tangental bodes n terms of the quermassntegrals: Theorem 3.1 (Favard [5], [19, p. 367]). Let K, E K0 n, E K, and let p {0,..., n 1}. Then K s a p-tangental body of E f and only f W 0 (K; E) = W 1 (K; E) = = W n p (K; E). Ths property may be rephrased n terms of the roots as follows where we exclude the trval case of 0-tangental bodes. Proposton 3.1. Let K, E K n 0, E K, let γ 1,..., γ n be the roots of f K;E (z) and let p {1,..., n 1}. Then K s a p-tangental body of E f and only f there exst constants α j, j = 1,..., p, wth 0 < α p < 1 such that the γ s are the roots of the polynomal g α,p (z) = (z + 1) n p 1 k=0 ( n k) αp k z n k. Proof. If K K0 n s a p-tangental body of E, Theorem 3.1 asserts that W 0 (K; E) = W (K; E), for all = 1,..., n p, and so we may wrte n p ( ) f K;E (z) = W 0 (K; E) n z W (K; E) + W =0 =n p+1 0 (K; E) z [ p 1 ( ) ( n = W 0 (K; E) (z + 1) n 1 W ) ] n k(k; E) z n k. k W 0 (K; E) k=0 Thus, settng α p k = 1 W n k (K; E)/W 0 (K; E), k = 0,..., p 1, all γ s are roots of the polynomal g α,p (z). Snce α p = 1 vol(e)/vol(k) we have by our assumpton E K that 0 < α p < 1. Conversely, we now assume that there exst p constants α j, j = 1,..., p, 0 < α p < 1, such that the γ s are the roots of g α,p (z). Snce α p < 1 the polynomal g α,p (z) s of degree n. Hence the γ s are roots of the polynomal p 1 (z + 1) n α p k z n k = k k=0 n p =0 z + =n p+1 (1 α p+ n )z

10 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE as well as of f K;E (z) = n =0 ( n ) W (K; E)z. So both polynomals have the same coeffcents up to a constant c, say. Snce α p < 1 we have c > 0 and by comparng the coeffcents we get W 0 (K; E) = W 1 (K; E) = = W n p (K; E) = c. Theorem 3.1 now ensures that K s a p-tangental body of E. We note that the constants α j s n the theorem above actually satsfy 0 < α 1 α p < 1, whch follows drectly from (3.1) snce r(k; E) = 1. In the partcular case of cap-bodes (1-tangental bodes) the characterzaton s more explct. Corollary 3.1. Let K, E K n 0, E K, and let γ 1,..., γ n be the roots of f K;E (z). Then K s a cap-body of E f and only f there exsts α (0, 1) such that (1/α) 1/n (1 + 1/γ ), = 1,..., n, are the n roots of unty. Proof. Proposton 3.1 states that K s a 1-tangental body of E f and only f there exsts α (0, 1) such that γ k, k = 1,..., n, satsfy the equaton (z + 1) n = αz n. Snce W 0 (K, E) > 0 no root of f K;E (z) can be zero and so we have equvalently (1 + 1/z) n = α. Therefore, K s a 1-tangental body of E f and only f (3.2) 1 + 1 γ k = α 1/n e 2π(k 1) n, for k = 1,..., n and some α (0, 1). Remark 3.1. Obvously, wth (3.2) we can explctly determne the roots of the Stener polynomal of a cap-body K of E, namely 2π(k 1) 1 + β cos n β sn 2π(k 1) n (3.3) γ k = ( ), k = 1,..., n, 1 + β β 2 cos 2π(k 1) n wth β = α 1/n = ( 1 vol(e)/vol(k) ) 1/n. Fgure 2 shows n a thcker lne all roots n R(3, B 3 ) of Stener polynomals of cap-bodes of B 3 (cf. Fgure 1). 0.8 0.6 0.4 0.2-2 -1.5-1 -0.5 Fgure 2. Complex numbers z C + whch are roots of f K;B3 (z) for any cap-body K of B 3.

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 11 II. On constant wdth sets. Next we deal wth the famly of constant wdth sets, for whch we get a symmetry condton on the roots of ther Stener polynomals. A convex body K s called a constant wdth set f t has the same breath (say b) n any drecton,.e., the dstance between any two parallel supportng hyperplanes s b. In order to establsh ths symmetry condton we need the followng general result on polynomals. Lemma 3.1. Let f(z) = a 0 + a 1 z + + a n z n be a real polynomal wth a n 0 and let b R. Then k (3.4) a n k = ( 1) b k a n n k =0 for k = 0, 1,..., n, f and only f all the roots of f(z) are symmetrc wth respect to b/2,.e., γ s a root f and only f b γ s a root. Proof. The polynomal f( b z) s gven by f( b z) = a k ( b z) k = (3.5) = = k=0 k=0 k=0 = ( 1) n k=0 l=0 ( ) k )b k z ( k a k ( 1) k k=0 =0 ( ( ) m a m ( 1) )b m m k z k k m=k ( n k ( ) n l a n l ( 1) )b n l n k l z k k l=0 ( n k ( ) n l a n l ( 1) )b l n k l z k. k The leadng coeffcent s ( 1) n a n and thus the two polynomals f(z) and f( b z) have the same roots f and only f f(z) = ( 1) n f( b z),.e., f and only f the relatons (3.4) hold for all k = 0, 1,..., n. As a drect consequence we get the followng statement for Stener polynomals. Theorem 3.2. Let b > 0. A convex body K K n verfes the relatons k ( ) k (3.6) W n k (K; E) = ( 1) b k W n (K; E) =0 for k = 0, 1,..., n, f and only f all the roots of ts Stener polynomal f K;E (z) are symmetrc wth respect to b/2. Proof. Snce ( )( ) ( )( ) n k n n = for k, k n k the result follows mmedately from Lemma 3.1.

12 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE Observe that the equatons (3.6) are not ndependent, snce they are equvalent to the (n + 1)/2 relatons obtaned for only odd values of k. Notce also that symmetry wth respect to b/2 0 n Theorem 3.2 s not possble. Indeed, for b = 0 we would get that all odd quermassntegrals have to vansh whch s not possble, and n the case b > 0 t would mply that W j (K; E) = 0 for all j = 1,..., n, agan not possble. When E = B n, t s known (see [4]) that bodes of constant wdth fulfll (3.6), whch shows Proposton 1.2. In partcular, f n = 3 the roots of the Stener polynomal of the constant wdth set K are γ 1 = b 2, γ 2, γ 3 = b 2 ± 3 2 b 2 3W 1(K), π whch are real, snce πb 2 3W 1 (K) (cf. (2.1); see also [8, Example 3]). Notce that f (3.6) holds and the dmenson s odd then b/2 s a root of f K;E (z), whereas f n s even and b/2 s a root of f K;E (z), then t has to be at least a double root. III. Balls and sausages. Fnally we consder for K balls (more generally the gauge body E) and sausages. The gauge body E can be seen both as a very specal tangental body (t s a 0-tangental body of E) and as a very partcular case of a body satsfyng (3.6). It can be also characterzed n a more general way: Proposton 3.2. Let K K n, let γ, = 1,..., n, be the roots of f K;E (z), and let a > 0. Then Re(γ ) = a > 0 for all = 1,..., n f and only f K = ae (up to translatons). Proof. If K = ae, then f K;E (z) = W n (K; E)(z + a) n, and hence t has an n-fold real root γ = a. So let us assume Re(γ ) = a for = 1,..., n. From f K;E (z) = W n (K; E) n =1 (z γ ) we get the relatons ma = Re(γ ) = γ = n W n 1(K; E) W n (K; E), =1 =1 =1 n =1 γ = ( 1) n W 0(K; E) W n (K; E), for some 1 m n. We note that we can exclude the case m = 0, because a > 0 mples K K0 n and thus W n 1(K; E) 0. Hence W 0 (K; E) n (3.7) W n (K; E) = γ a n W n 1 (K; E) n = W n 1(K; E) n m W n (K; E) W n (K; E) n,.e., W n (K; E) n 1 W 0 (K; E) W n 1 (K; E) n. However Mnkowsk s frst nequalty states that W n (K; E) n 1 W 0 (K; E) W n 1 (K; E) n, where equalty holds for K, E K0 n f and only f K and E are homothetc (see e.g. [19, p. 317]). Thus there exst α > 0 and t R n such that K = α E + t. From

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 13 a n = W 0 (K; E)/W n (K; E) we get α = a, and, n partcular, m = n and γ = a for all = 1,..., n. Theorem 1.3 wll be a drect consequence of the followng result for whch we denote by f (n 1) K;B n (z) the (n 1)-st dervatve of f K;Bn (z). Theorem 3.3. Let K K n. Then r(k) s root of f (n 1) K;B n (z) and all ts 2-dmensonal projectons onto any 2-dmensonal lnear subspace have nradus r(k) f and only K s a sausage wth nradus r(k). Proof of Theorem 3.3. Frst we need some addtonal notaton. The set of all k-dmensonal lnear subspaces of R n wll be denoted by L n k, and for L L n k and K Kn, K L denotes the orthogonal projecton of K onto L. Fnally, W (k) denotes the -th quermassntegral computed n a k-dmensonal affne subspace. The quermassntegrals of a sausage S = L+rB n, where L s a lne segment wth length l, are gven by [ ] n k W k (S) = r n k 1 n vol n 1(B n 1 ) l + vol(b n ) r. Here vol j (M) denotes the j-dmensonal volume of M R j. Thus t can be easly checked that r = r(s) s an (n 1)-fold root of f S;Bn (z) and so a root of ts (n 1)-st dervatve. So we assume that r = r(k) s root of f (n 1) K;B n (z),.e., (3.8) W n 2 (K) 2rW n 1 (K)+ r 2 W n (K) = 0, and let r(k L; B n L) = r(k) for all projectons onto 2-dmensonal planes L L n 2. Let k {1,..., n}. Kubota s ntegral recurson formula states that for any = 0,..., k, the (n k + )-th quermassntegral W n k+ (K) can be expressed as (3.9) W n k+ (K) = vol(b n) W (k) (K L) dσ(l), vol k (B k ) where σ(l) s the Haar measure on the set L n k such that σ(ln k ) = 1 (see e.g. [19, p. 295, (5.3.27)]). Let k = 2 and L L n 2. The Stener polynomal of K L as a 2-dmensonal set s gven by f K L;B2 (z) = 2 ( 2 ) (2) =0 W (K L)z. Applyng (3.9) we get f K L;B2 ( r) dσ(l)= W (2) 0 (K L) dσ(l) 2r W (2) 1 (K L) dσ(l) L n 2 L n 2 L n 2 + r 2 W (2) 2 (K L) dσ(l) L n 2 L n k = vol 2(B 2 )[ Wn 2 (K) 2rW n 1 (K)+ r 2 W n (K) ] = 0 vol(b n )

14 MARTIN HENK AND MARÍA A. HERNÁNDEZ CIFRE by (3.8). Snce r = r(k L; B n L) for all L L n 2, Bonnesen s nequalty (cf. (1.4)) states that (3.10) f K L;B2 ( r) = W (2) 0 (K L) 2W(2) 1 (K L)r + W(2) 2 (K L)r2 0, wth equalty f and only f K L s a 2-dmensonal sausage wth nradus r. Thus, from f K L;B2 ( r) dσ(l) = 0 L n 2 and (3.10) we conclude f K L;B2 ( r) = 0 for any L L n 2 whch mples that K L s a 2-dmensonal sausage wth nradus r for any L L n 2. By [19, Lemma 3.2.6] ths s equvalent to K beng an n-dmensonal sausage wth nradus r. References [1] W. Blaschke, Vorlesungen über Integralgeometre, Deutscher Verlag der Wssenschaften, Berln, 1995, 3rd ed. (1st ed.: 1949). [2] G. Bol, Bewes ener Vermutung von H. Mnkowsk, Abh. Math. Sem. Hansschen Unv. 15 (1943), 37 56. [3] T. Bonnesen, Les problèmes des sopérmètres et des sépphanes, Collecton de monographes sur la théore des fonctons, Gauther-Vllars, Pars 1929. [4] A. Dnghas, Verallgemenerung enes Blaschkeschen Satzes über konvexe Körper konstanter Brete, Rev. Math. Unon Interbalkan. 3 (1940), 17 20. [5] J. Favard, Sur les corps convexes, J. Math. Pures Appl. 12 (9) (1933), 219 282. [6] P. M. Gruber, Convex and Dscrete Geometry. Sprnger, Berln Hedelberg, 2007. [7] M. Henk, M. A. Hernández Cfre, Notes on the roots of Stener polynomals, Rev. Mat. Iberoamercana 24 (2) (2008), 631 644. [8] M. A. Hernández Cfre, E. Saorín, On the roots of the Stener polynomal of a 3- dmensonal convex body, Adv. Geom. 7 (2007), 275 294. [9] M. A. Hernández Cfre, E. Saorín, Some geometrc propertes of the roots of the Stener polynomal, Rend. Crc. Mat. Palermo 77 (II) (2006), 319 332. [10] M. A. Hernández Cfre, E. Saorín, How to make quermassntegrals dfferentable: solvng a problem by Hadwger, Submtted. [11] M. Jetter, Bounds on the roots of the Stener polynomal, to appear n Adv. Geom. [12] O. M. Katkova, A. M. Vshnyakova, A suffcent condton for a polynomal to be stable, J. Math. Anal. Appl. 347 (1) (2008), 81 89. [13] V. Katsnelson, On H. Weyl and J. Stener polynomals, Complex Anal. Oper. Theory 3 (1) (2009), 147 220. [14] M. Marden, Geometry of polynomals, Mathematcal Surveys, No. 3, Amercan Mathematcal Socety, Provdence, R.I., 1966, 2nd ed. [15] G. Matheron, La formule de Stener pour les érosons, J. Appl. Prob. 15 (1978), 126 135. [16] Y. Y. Ne, X. K. Xe, New crtera for polynomal stablty, IMA J. Math. Control Inform. 4 (1) (1987), 1 12. [17] J. R. Sangwne-Yager, Bonnesen-style nequaltes for Mnkowsk relatve geometry. Trans. Amer. Math. Soc. 307 (1) (1988), 373 382. [18] J. R. Sangwne-Yager, Mxed volumes. In: Handbook of Convex Geometry (P. M. Gruber and J. M. Wlls eds.), North-Holland, Amsterdam, 1993, 43 71. [19] R. Schneder, Convex Bodes: The Brunn-Mnkowsk Theory, Cambrdge Unversty Press, Cambrdge, 1993.

ON THE LOCATION OF ROOTS OF STEINER POLYNOMIALS 15 [20] J. Stener, Über parallele Flächen, Monatsber. Preuss. Akad. Wss. (1840), 114 118, [Ges. Werke, Vol II (Remer, Berln, 1882) 245 308]. [21] B. Tesser, Bonnesen-type nequaltes n algebrac geometry I. Introducton to the problem. Semnar on Dfferental Geometry, Prnceton Unv. Press, Prnceton, N. J., 1982, 85 105. Fakultät für Mathematk, Otto-von-Guercke Unverstät Magdeburg, Unverstätsplatz 2, D-39106-Magdeburg, Germany E-mal address: martn.henk@ovgu.de Departamento de Matemátcas, Unversdad de Murca, Campus de Espnardo, 30100-Murca, Span E-mal address: mhcfre@um.es