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A strengthening of the Blaschke-Santaló inequality for $o$-symmetric planar convex bodies

Károly J. Böröczky, Endre Makai

Abstract

We verify the inequality $$ \frac{|K|}{|E|}+\frac{|K^*|}{|E^*|}\leq 2 $$ for any $o$-symmetric convex body $K\subset\R^2$ where $E$ is either the John ellipse of maximal area contained in $K$ or the minimal area Löwner ellipse containing $K$. The analogous estimate may not hold if $K$ is a planar but the assumption of $o$-symmetry is dropped, or if $K$ is $o$-symmetric convex body in $\R^n$ for $n\geq 3$. Our new inequality strengthens the Blaschke-Santaló inequality for $o$-symmetric convex bodies $K\subset\R^2$ with an error term of optimal order.

A strengthening of the Blaschke-Santaló inequality for $o$-symmetric planar convex bodies

Abstract

We verify the inequality for any -symmetric convex body where is either the John ellipse of maximal area contained in or the minimal area Löwner ellipse containing . The analogous estimate may not hold if is a planar but the assumption of -symmetry is dropped, or if is -symmetric convex body in for . Our new inequality strengthens the Blaschke-Santaló inequality for -symmetric convex bodies with an error term of optimal order.
Paper Structure (3 sections, 6 theorems, 66 equations)

This paper contains 3 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Area sum within a convex cone
  3. Proof of Theorem \ref{['thm-volume-sum']} and Theorem \ref{['BS-stab']}

Key Result

Theorem A

If $K\subset\mathbb R^n$ is an $o$-symmetric convex body, then with equality if and only if $K$ is an ellipsoid.

Theorems & Definitions (19)

  • Theorem A: Blaschke-Santaló inequality
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Example 1.4: The order $\sqrt{\varepsilon}$ of the error term in \ref{['eq-BS-stab']} is optimal
  • Proposition 2.1
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • ...and 9 more