An estimate for the measure of nonconvexity in the $L^p$-space

The measure $\alpha(A)$ of nonconvexity for a bounded subset $A$ of a normed linear space $L$ is the Hausdorff distance between $A$ and its convex hull co $A$. It is proved that if $L$ is an $L^p$-space, then $\alpha(A)\le d(A)/2^{t_p}$, where $d(A)$ is the diameter of $A$ and $t_p=\min\{1/p,1-1/p\}$, $1\le p\le\infty$.Furthermore, this estimate is sharp.