Shadow Systems and Volumes of Polar Convex Bodies

Meyer, Mathieu; Reisner, Shlomo

doi:10.1112/s0025579300000061

Cited by 57 publications

(60 citation statements)

References 17 publications

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“…, n where S i denotes the orthogonal symmetry with respect to the coordinate hyperplane {x ∈ R n ; x i = 0}. The conjecture was proved by Reisner [21] for zonoids (see also [8] for a simple proof) and by Meyer and Reisner [17] for polytopes with less than n + 3 vertices. For convex bodies close, in Banach-Mazur distance, to the cube (resp.…”

Section: K| |Kmentioning

confidence: 97%

The volume product of convex bodies with many hyperplane symmetries

Barthe¹,

Fradelizi²

2013

ajm

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Mahler's conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflection groups, namely direct products of the isometry groups of regular polytopes.

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Section: K| |Kmentioning

confidence: 97%

The volume product of convex bodies with many hyperplane symmetries

Barthe¹,

Fradelizi²

2013

ajm

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“…is a convex function of t. In [MR2], Reisner and the second author generalized this result to the non-symmetric case and studied the equality case. The following proposition is the key tool in the proof of Theorem 1:…”

Section: The Toolsmentioning

confidence: 99%

“…We shall need the following easy lemma (see, for example, Lemma 11 in [MR2]), where we use the fact that inequalities (1) and (2) hold in dimension 2, by the original Mahler's result. Lemma 1.…”

Section: The Toolsmentioning

confidence: 99%

“…Several special cases of the (mostly) symmetric case of the conjecture can be found in [SR,R1,GMR,M1,R2,NPRZ,KR,FGMR,RSW,GM]. Not many special cases in which (1) is true seem to be known, one such is the case of convex bodies having hyperplane symmetries which fix only one common point [BF]; another one is the n dimensional polytopes with at most n + 3 vertices (or facets) proved in [MR2]. The proof of this last result is based on the method of shadow systems.…”

Section: Introductionmentioning

confidence: 99%

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An Application of Shadow Systems to Mahler’s Conjecture

Fradelizi

Meyer

Zvavitch

2012

Discrete Comput Geom

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Abstract. We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product P(K) = min z∈int(K) |K|||K z |, where K ⊂ R n is a convex body and1 for all x ∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K ⊂ R 3 is the convex hull of two 2-dimensional convex bodies, then P(K) P(∆ 3 ), where ∆ 3 is a 3-dimensional simplex, thus confirming the 3-dimensional case of Mahler conjecture, for this class of bodies. A similar result is provided for the symmetric case, where we prove that if K ⊂ R 3 is symmetric and the convex hull of two 2-dimensional convex bodies, then P(K) P(B

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“…For n = 2, Mahler himself proved this inequality in 1939 (see, e.g., [5,6,14] for references) and Meyer [18] obtained the equality conditions in 1991. Recently, Meyer and Reisner [19] have proved inequality (1.2) for polytopes with at most n + 3 vertices. Very recently, Kim and Reisner [10] proved that the simplex is a strict local minimum for the Mahler volume in the Banach-Mazur space of n-dimensional convex bodies.…”

Section: Introductionmentioning

confidence: 99%