Bezout Inequality for Mixed Volumes

Soprunov, Ivan; Zvavitch, Artem

doi:10.1093/imrn/rnv390

Cited by 18 publications

(36 citation statements)

References 15 publications

(15 reference statements)

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“…191] for the definition of zonoids.) Indeed, similar to the proof of [SZ,Theorem 5.6], it is enough to show that (6.4) holds when L 1 , . .…”

Section: Weakly Decomposable Bodiesmentioning

confidence: 79%

“…The following theorem, which is the main result of this section, is a generalization of both facts that decomposable convex bodies and polytopes that are not simplices do not satisfy the Bezout inequality (5.1) for any 2 ≤ r ≤ n, see Theorem 4.1 and [SZ,Theorem 3.3].…”

Section: Weakly Decomposable Bodiesmentioning

confidence: 99%

“…. , L r and K in R n , see [SZ,Theorem 5.7]. Since then, several new results on estimating the constant c n,r have been obtained, see [AFO, BGL, Xi].…”

Section: Weakly Decomposable Bodiesmentioning

confidence: 99%

See 2 more Smart Citations

Wulff shapes and a characterization of simplices via a Bezout type inequality

Saroglou

Soprunov

Zvavitch

2019

Advances in Mathematics

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Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumesWe show that the above inequality characterizes simplices, i.e. if K is a convex body satisfying the inequality for all convex bodies L1, . . . , Ln ⊂ R n , then K must be an n -dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest.In addition, we study the Bezout inequality for mixed volumes introduced in [SZ]. We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in [SZ] characterizes weakly indecomposable convex bodies.2010 Mathematics Subject Classification. Primary 52A20, 52A39, 52A40, 52B11.

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“…191] for the definition of zonoids.) Indeed, similar to the proof of [SZ,Theorem 5.6], it is enough to show that (6.4) holds when L 1 , . .…”

Section: Weakly Decomposable Bodiesmentioning

confidence: 79%

Section: Weakly Decomposable Bodiesmentioning

confidence: 99%

See 1 more Smart Citation

Wulff shapes and a characterization of simplices via a Bezout type inequality

Saroglou

Soprunov

Zvavitch

2019

Advances in Mathematics

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“…Induction shows that (5.23) holds true with c r 2 2 r−1 −1 . Let us finally mention that Soprunov and Zvavitch have observed in [24] that if A = ∆ is an n-dimensional simplex then (5.23) holds true with constant 1, and they conjecture that if a convex body A in R n satisfies (5.23) with constant 1 for all r and all K 1 , . .…”

Section: Inequalities About Mixed Volumesmentioning

confidence: 97%

Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

Brazitikos

Giannopoulos

Liakopoulos

2018

Advances in Geometry

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The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality.

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“…, K r then ∆ is necessarily a simplex (see [SZ,Conjecture 1.2]). It was proved that ∆ has to be indecomposable (see [SZ,Theorem 3.3]) which, in particular, confirms the conjecture in dimension n = 2. In the present paper we prove this conjecture for the class of convex polytopes.…”

Section: Introductionmentioning

confidence: 99%

Characterization of simplices via the Bezout inequality for mixed volumes

Saroglou

Soprunov

Zvavitch

2016

Proc. Amer. Math. Soc.

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Abstract. We consider the following Bezout inequality for mixed volumes:It was shown previously that the inequality is true for any n -dimensional simplex ∆ and any convex bodies K1, . . . , Kr in R n . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies K1, . . . , Kr in R n . In this paper we prove that this is indeed the case if we assume that ∆ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex n -polytopes. In addition, we show that if a body ∆ satisfies the Bezout inequality for all bodies K1, . . . , Kr then the boundary of ∆ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

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Bezout Inequality for Mixed Volumes

Cited by 18 publications

References 15 publications

Wulff shapes and a characterization of simplices via a Bezout type inequality

Wulff shapes and a characterization of simplices via a Bezout type inequality

Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

Characterization of simplices via the Bezout inequality for mixed volumes

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