Continuity and affine invariance of measures of convexity

Schneider asks [8] whether there exists an affine invariant and continuous measure of convexity on the space Cof compact subsets of a finite-dimensional Euclidean space. This question is considered still open. We provide a negative answer to the previous question: any affine invariant measure of convexity on Ccannot be continuous in terms of the Hausdorff metric. We also show that some weaker form of continuity can still be retained for affine invariant measures of convexity by showing that the Schneider’s measure proposed in [7] is lower semi continuous.