Convex Geometry II. Distances and Measures, Approximation, Comparison and Symmetrisation

A first article [AF 219] dealing with Convex Geometry covered the main definitions and properties of convex sets and, more broadly, star sets, as well as fundamental theorems concerning them.

This second article deals with more advanced notions, which are nonetheless extremely useful for practical applications (data analysis, image analysis, shape analysis...). Convex sets need to be measured (volumes, surface areas, diameters...), hence the need for appropriate mathematical "measures" (intrinsic volumes) providing geometric quantities commonly referred to as size descriptors. They must also be approximated and compared, hence the need for distance functions (Pompeiu and Hausdorff distance, Asplund distance...). The existence of geometric inequalities linking these geometric quantities also enables the construction of shape descriptors. Questions concerning the continuity of geometric quantities and the convergence of sequences of convex or star-shaped subsets are not solely theoretical, but arise (or must arise) in many practical cases involving problems of comparison, approximation and symmetrization.