A note on the affine-invariant plank problem

TL;DR

This paper proves a conjecture related to the affine-invariant plank problem, establishing bounds on the sum of relative widths of planks covering convex sets, under certain convexity assumptions and dimensional conditions.

Contribution

It provides a short proof of Bang's conjecture for the affine-invariant plank problem under a convexity assumption and extends the result to bounds involving the dimension of projections.

Findings

Proved Bang's conjecture under the assumption that the remaining set after covering is convex.

Established that the sum of relative widths is at least 1/d, where d is the dimension of the projection of C.

Provided a simplified proof technique for the affine-invariant plank problem.

Abstract

Suppose that $C$ is a bounded, convex subset of $\mathbb{R}^n$, and that $P_1, \dots, P_k$ are planks which cover $C$ in respective directions $v_1, \dots, v_k$ and with widths $w_1, \dots, w_k$. In 1951, Bang conjectured that the sum of relative widths $$\sum_{i=1}^k \frac{w_i}{w_{v_i}(C)} \geq 1, $$ generalizing a previous conjecture of Tarski. Here, $w_{v_i}(C)$ is the width of $C$ in the direction $v_i$. In this note we give a short proof of this conjecture under the assumption that, for every $m$ with $1 \leq m \leq k$, $ C \setminus \bigcup_{i = 1}^m P_i $ is a convex set. In addition, we prove that if the projection of $C$ onto the vector space spanned by the normal vectors of the planks has dimension $d$, then the above sum of relative widths is at least $1/d$.