Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound

TL;DR

This paper extends a stability result for barriers of convex bodies in n-dimensional space, showing that near-minimal surface area barriers have orientations close to symmetrized bodies, and introduces the concept of weak barriers.

Contribution

It generalizes a known 2D stability result to higher dimensions and introduces weak barriers that focus on orientation rather than position.

Findings

Barriers with surface area close to Jones' bound have orientations close to symmetrized bodies.

The concept of weak barriers is introduced to analyze orientation without positional information.

Quantitative stability results are obtained for weak barriers in all dimensions.

Abstract

Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space. A set $B$ is called a barrier (or an `opaque set') for $K$ if every line that intersects $K$, also intersects $B$. Although this concept was introduced more than a century ago, the barrier with minimal surface area for a given set $K$ is still unknown, even in the two-dimensional case. A classical lower bound by Jones states that the surface area $S(B)$ of a sufficiently regular barrier $B$ is at least $S(\partial K)/2$, half the surface area of the boundary of $K$. We will extend a known stability version for $n=2$ to arbitrary dimensions: if $S(B)-S(\partial K)/2$ is small, then the orientation measure of $B$ is close to the surface area measure of a symmetrization of $K$. For instance, if $K$ is the unit cube in 3D, most of the points of a barrier with surface area close to $3$ must have almost axis parallel normals. One of the main contributions of the paper is the new concept of weak barriers, which only encodes orientation information of a barrier, disregarding the relative positions of its parts. We characterize weak barriers geometrically in terms of the convexification of $B$. Convex geometric tools then allow one to quantify the above mentioned stability for weak barriers in all dimensions.