A note on 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 $1/d$.