The Inverse Kakeya Problem

TL;DR

This paper proves that the largest convex shape that can be oriented inside a convex body is always the inscribed ball, with uniqueness except in perimeter maximization in 2D.

Contribution

It establishes the optimality and uniqueness of the inscribed ball as the largest convex shape in any orientation within a convex body.

Findings

The largest inscribed ball maximizes volume and surface area in any orientation.

Uniqueness of the inscribed ball as the optimal shape, except for perimeter in 2D.

The result extends to both volume and surface area maximization.

Abstract

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest volume" and when it means "largest surface area". The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.