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 $$\sum _{i=1}^k\frac{w_i}{w_{v_i}(C)}\ge 1,$$ 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 $1\le m\le k$, $C\setminus {\bigcup }_{i=1}^m{P}_i$ 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 $1/d$.

8 pages, 1 figure