Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909040259366912 |
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| author | Kiderlen, Markus |
| author_facet | Kiderlen, Markus |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13449 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound Kiderlen, Markus Metric Geometry 52A20 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. |
| title | Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound |
| topic | Metric Geometry 52A20 |
| url | https://arxiv.org/abs/2605.13449 |