Mahler's conjecture and curvature

Shlomo Reisner; Carsten Schütt; Elisabeth M. Werner

doi:10.1093/imrn/rnr003

Mahler's conjecture and curvature

Shlomo Reisner, Carsten Schütt, Elisabeth M. Werner

Institute for Mathematics and its Applications

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

Let K be a convex body in ℝⁿ with Santaló point at 0. We show that if K has a point on the boundary with positive generalized Gauß curvature, then the volume product |K||K°|is not minimal. This means that a body with minimal volume product has a Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

Original language	English (US)
Pages (from-to)	1-16
Number of pages	16
Journal	International Mathematics Research Notices
Volume	2012
Issue number	1
DOIs	https://doi.org/10.1093/imrn/rnr003
State	Published - Jan 1 2012

Bibliographical note

Funding Information:
This work was partially supported by an NSF grant, a FRG-NSF grant and a BSF grant (to E.W.).

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10.1093/imrn/rnr003

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abstract = "Let K be a convex body in ℝn with Santal{\'o} point at 0. We show that if K has a point on the boundary with positive generalized Gau{\ss} curvature, then the volume product |K||K°|is not minimal. This means that a body with minimal volume product has a Gau{\ss} curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.",

author = "Shlomo Reisner and Carsten Sch{\"u}tt and Werner, \{Elisabeth M.\}",

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N2 - Let K be a convex body in ℝn with Santaló point at 0. We show that if K has a point on the boundary with positive generalized Gauß curvature, then the volume product |K||K°|is not minimal. This means that a body with minimal volume product has a Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

AB - Let K be a convex body in ℝn with Santaló point at 0. We show that if K has a point on the boundary with positive generalized Gauß curvature, then the volume product |K||K°|is not minimal. This means that a body with minimal volume product has a Gauß curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.

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Mahler's conjecture and curvature

Abstract

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