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

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.