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John-type decompositions for affinely-optimal positions of convex bodies

Florian Grundbacher, Tomasz Kobos

Abstract

Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describes$\unicode{x2014}$and in the Euclidean case fully characterizes$\unicode{x2014}$the optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains $r L_1 + c \subseteq K \subseteq R L_2 + d$ for convex bodies $K, L_1, L_2 \subseteq \mathbb{R}^n$, translation vectors $c,d \in \mathbb{R}^n$, and reals $r,R > 0$ such that the ratio $\frac{R}{r}$ cannot be decreased by linearly transforming $K$. We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball.

John-type decompositions for affinely-optimal positions of convex bodies

Abstract

Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of convex bodies contained in other convex bodies, where the John decomposition describesand in the Euclidean case fully characterizesthe optimal positions. For many other such problems, however, no general optimality conditions are known. To address this, we generalize an approach of O. B. Ader to obtain a John-type decomposition as a necessary condition for affinely-optimal containment chains, i.e., chains for convex bodies , translation vectors , and reals such that the ratio cannot be decreased by linearly transforming . We again obtain sufficiency for optimality when ellipsoids are involved, and show how optimality conditions for various problems follow from our result. Our main applications concern the Banach-Mazur distance, where we provide necessary optimality conditions in the general case and a full characterization in the Euclidean case. Finally, we derive several consequences of these optimality conditions related to the Banach-Mazur distance to the Euclidean ball.
Paper Structure (6 sections, 19 theorems, 136 equations, 1 figure)

This paper contains 6 sections, 19 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction and Results
  2. Existence of an Ader Decomposition in the Affinely-Optimal Position
  3. Applications for Decomposition-Type Results
  4. The Mean Ellipsoid Theorem
  5. Characterization of Affinely-Optimal Positions Involving Ellipsoids
  6. Applications for the Banach--Mazur distance to the Euclidean ball

Key Result

Theorem 1.1

Let $K \subseteq \mathbb{R}^n$ be a convex body such that $\mathbb{B}^n \subseteq K$. Then $\mathbb{B}^n$ is the unique ellipsoid of maximal volume contained in $K$ if and only if there exist contact points $u^1, \ldots, u^N \in \mathop{\mathrm{bd}}\nolimits(K) \cap \mathop{\mathrm{bd}}\nolimits(\ma In this case, $\sum_{i=1}^N \lambda_i = n$ and there exists a choice with $N \leq \frac{n(n+3)}{2}$

Figures (1)

  • Figure 1: An example showing that the "natural candidate" for the outer mean ellipsoid may not satisfy the desired containment relation: $E_0 + c_0$ (blue, dashed), $E_1 + c_1$ (red, dotted), $E_\lambda + c_\lambda$ (purple, solid), $E_\lambda + x$ (black, dash-dotted).

Theorems & Definitions (37)

  • Theorem 1.1: John Ellipsoid Theorem
  • Theorem 1.2
  • Theorem 1.3: John decomposition in the general case
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:generaldecomp']}.
  • ...and 27 more