New affine measures of symmetry for convex bodies

Mathieu Meyer; Carsten Schütt; Elisabeth M. Werner

doi:10.1016/j.aim.2011.08.002

New affine measures of symmetry for convex bodies

Mathieu Meyer, Carsten Schütt, Elisabeth M. Werner

Institute for Mathematics and its Applications

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

11 Scopus citations

Abstract

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with "enough symmetries". To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are "far apart". In particular, we give an example of a convex body whose centroid and Santaló point are "far apart".

Original language	English (US)
Pages (from-to)	2920-2942
Number of pages	23
Journal	Advances in Mathematics
Volume	228
Issue number	5
DOIs	https://doi.org/10.1016/j.aim.2011.08.002
State	Published - Dec 1 2011

Bibliographical note

Funding Information:
✩ Partially supported by an NSF grant, an FRG-NSF grant and a BSF grant. * Corresponding author at: Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, USA. E-mail addresses: mathieu.meyer@univ-mlv.fr (M. Meyer), schuett@math.uni-kiel.de (C. Schütt), elisabeth.werner@case.edu (E.M. Werner).

Keywords

Convex bodies
Measures of symmetry
Santaló point

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10.1016/j.aim.2011.08.002

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title = "New affine measures of symmetry for convex bodies",

abstract = "Gr{\"u}nbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with {"}enough symmetries{"}. To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are {"}far apart{"}. In particular, we give an example of a convex body whose centroid and Santal{\'o} point are {"}far apart{"}.",

keywords = "Convex bodies, Measures of symmetry, Santal{\'o} point",

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

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AU - Schütt, Carsten

AU - Werner, Elisabeth M.

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AB - Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with "enough symmetries". To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are "far apart". In particular, we give an example of a convex body whose centroid and Santaló point are "far apart".

KW - Convex bodies

KW - Measures of symmetry

KW - Santaló point

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New affine measures of symmetry for convex bodies

Abstract

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