On the relation between $\alpha$-sets and weakly convex sets

The known relation between $\alpha$-sets and Vial weakly convex sets in Euclidean spaces of dimension greater than two is strengthened. Namely, in the formula describing the relationship between nonconvexity measures for $\alpha$-sets and weakly convex sets, the double Chebyshev radius is replaced by the diameter of the set with a coefficient. However, in the two-dimensional space the corresponding estimate is expressed in terms of the set diameter without a coefficient and is more accurate. In this connection, the question of the possibility of further refinement of the estimate for the nonconvexity degree $\alpha$ in terms of the weak convexity parameter $R$ and the set diameter in Euclidean spaces of dimension greater than two remains open.