  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
$$\sum_{i=1}^k \frac{w_i}{w_{v_i}(C)} \geq 1, $$ 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 $$ \sum_{i=1}^k \frac{w_i}{w_{v_i}(C)} \geq \frac{1}{d}. $$
