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Tightening Inequalities on Volume Extremal $k$-Ellipsoids Using Asymmetry Measures

René Brandenberg, Florian Grundbacher

TL;DR

This work studies two affine-invariant problems for k-dimensional ellipsoids tied to John and Loewner ellipsoids: lower bounds on the volume of outer k-ellipsoids containing projections and upper bounds on the volume of inner k-ellipsoids contained in sections. A unifying approach via the John asymmetry $s_J(K)$ sharpens both bounds, with explicit equality characterizations and tightness across symmetric bodies, cross-polytopes, cubes, and simplices. In the planar case, the authors derive an exact, stronger bound for diameters in terms of $s_J(K)$, highlighting the role of asymmetry in extremal configurations. Extending beyond k-ellipsoids, they analyze affine optimization of width, circumradius, diameter, and inradius ratios, connecting these bounds to Banach-Mazur and Grünbaum distances and yielding sharp results for several classical convex bodies.

Abstract

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach-Mazur distance.

Tightening Inequalities on Volume Extremal $k$-Ellipsoids Using Asymmetry Measures

TL;DR

This work studies two affine-invariant problems for k-dimensional ellipsoids tied to John and Loewner ellipsoids: lower bounds on the volume of outer k-ellipsoids containing projections and upper bounds on the volume of inner k-ellipsoids contained in sections. A unifying approach via the John asymmetry sharpens both bounds, with explicit equality characterizations and tightness across symmetric bodies, cross-polytopes, cubes, and simplices. In the planar case, the authors derive an exact, stronger bound for diameters in terms of , highlighting the role of asymmetry in extremal configurations. Extending beyond k-ellipsoids, they analyze affine optimization of width, circumradius, diameter, and inradius ratios, connecting these bounds to Banach-Mazur and Grünbaum distances and yielding sharp results for several classical convex bodies.

Abstract

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach-Mazur distance.
Paper Structure (7 sections, 23 theorems, 176 equations, 1 figure)

This paper contains 7 sections, 23 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction and Notation
  2. Minimal Ellipsoids Containing Projections
  3. Maximal Ellipsoids Contained in Sections
  4. The Planar Case
  5. Affine Width-Circumradius- and Diameter-Inradius-Ratios
  6. Approximating Convex Bodies by Full-Dimensional Ellipsoids
  7. Technical Analysis of Functions

Key Result

Proposition 1.1

Let $k \in [n-1]$, $K \in \mathcal{K}^n$ with $\mathcal{E}_J(K) = \mathbb{B}^n$, and $E \subset K$ be a $k$-ellipsoid. Then with equality for the inscribed Euclidean $k$-balls of $k$-faces of regular simplices.

Figures (1)

  • Figure 1: The points defined in the proof of Theorem \ref{['thm:planar_diam']} for $s_J(K) = 1.6$ and $\xi = 1.7$. The solid (resp. dashed) circle has radius $1$ (resp. $\xi$). The dotted line consist of all points $v \in \mathbb{R}^2$ with $v_1 = - \frac{2}{\xi}$. One of the $u^i$ must lie on the minor circular arc connecting $\tilde{q}^1$ and $q^2$ (red).

Theorems & Definitions (51)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Example 2.3
  • ...and 41 more