Costa–Cover conjecture on the concavity of parallel volume

Let $A$ be a bounded measurable set in $cmathbb{R}^n$, let $B_2^n$ denote the Euclidean closed unit ball, and let $|\cdot|$ denote Lebesgue measure. Costa–Cover conjecture. The function

$$t \mapsto |A+tB_2^n|^{\frac{1}{n}}$$

is concave on $\mathbb{R}_+$. This conjecture is the parallel-volume analogue of the concavity of entropy power and concerns a Brunn–Minkowski-type property for Euclidean outer parallel sets; its resolution status is not specified in the source.