Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.08842

Explicit lower bounds for opaque sets of unit square and unit disc

Markus Kiderlen·Florian Pausinger
View original source ↗Read source document ↗Export metadata ↓

Abstract

Explicit lower bounds for the length of the shortest opaque set for the unit disc and the unit square in the Euclidean plane are derived. The results are based on an explicit application of the general method of Kawamura, Moriyama, Otachi and Pach. Employing a recent observation by Steinerberger on the possible orientations of straight barriers with length close to Jones' bound, we improve the bound for the unit square by more than a factor $3$. The bound for barriers of the unit disc is new and based on the idea that the free parameters in the general method from can be optimized due to the strong symmetry properties of the disc. Our approach illustrates both the power and the limitations of the method.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Markus Kiderlen, Florian Pausinger. 2025-12-10. Explicit lower bounds for opaque sets of unit square and unit disc. https://arxiv.org/abs/2509.08842

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An approximate counterexample to the Barker--Larman problem in dimension $4$

The Barker--Larman problem asks if a convex body $K \subseteq \mathbb R^n$ containing the Euclidean ball $\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $λ_0 = 4 \sqrt{3}π$ and any $N \in \mathbb N$, we obtain the existence of a family of convex bodies $K_{λ,N}$ with $λ\in (λ_0-r_N, λ_0 + r_N)$, such that the sections of $K_{λ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |λ- λ_0|^{N+1}$ of $λ$, while the difference between outradius and inradius of $K_{λ,N}$ is larger than $C |λ- λ_0|$. The bodies $K_{λ,N}$ are constructed via radial functions as \[ρ_{K_{λ,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(λ-λ_0)^n}{n!} φ_n(t) \right)^{-1},\] where $t \in [0,2π), λ\in \mathbb R$ and $φ_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \times 10^{-7}$.

math.MG↗
arXiv2026

On convex bodies with rotationally symmetric planar projections

Let $n\ge 3$ and let $K\subset\mathbb R^n$ be a convex body. For a two-dimensional linear subspace $P\subset\mathbb R^n$, let $K|P$ be the orthogonal projection of $K$ onto $P$. We prove that if for every two-dimensional subspace $P$, the planar convex body $K|P$ has $q$-fold rotational symmetry up to translation, then for $q\ge4$, this forces $K$ to be an Euclidean ball. The case $q=3$ is exceptional: non-spherical examples exist.

math.MG↗
arXiv2026

A Spectral Proof of Khachiyan's Ellipsoid Conjecture

For a convex body $K\subset\mathbb{R}^n$, let $w(K)$ denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace $H$ whose boundary passes through the center of the maximizing ellipsoid satisfies \[ w(K\cap H)\le\frac{\sqrt e}{2}\,w(K). \] The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.

math.MG↗