On the volume product of planar polar convex bodies-Lower estimates with stability

K. Böröczky, E. Makai, M. Meyer, S. Reisner

Research output: Contribution to journalArticlepeer-review

Abstract

Let K2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K S2 is a convex body, with o int K, then |K| · |K*| 27/4, with equality if and only if K is a triangle and o is its centroid. If K S2 is a convex body, then we have |K| · |[(K-K)/2)]*| a 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*|2 sin 2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K||K*| n2 sin2(π/n) to bodies with o int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

Original languageEnglish
Pages (from-to)159-198
Number of pages40
JournalStudia Scientiarum Mathematicarum Hungarica
Volume50
Issue number2
DOIs
StatePublished - 1 Jun 2013

Keywords

  • Banach-Mazur distance
  • Blaschke-Santaló inequality
  • lower estimates
  • Primary 52A40
  • reverse Blaschke-Santaló inequality
  • Santaló point
  • Secondary 52A38, 52A10
  • stability
  • volume product in the plane

ASJC Scopus subject areas

  • Mathematics (all)

Fingerprint

Dive into the research topics of 'On the volume product of planar polar convex bodies-Lower estimates with stability'. Together they form a unique fingerprint.

Cite this