Convex Bodies: Mixed Volumes and Inequalities

Abstract

We give a brief introduction into the theory of mixed volumes of convex bodies and discuss the inequalities involving volumes and mixed volumes: the Brunn–Minkowski, the Alexandrov–Fenchel, and the two Minkowski inequalities. Along the way we discuss the Steiner formula and the integral-geometric formulas, namely the proportionality of the average width to the total mean curvature and the formulas of Cauchy and Crofton. We also pay attention to the interplay between the discrete and the smooth, that is between convex polyhedra and convex hypersurfaces. The connections between the second Minkowski inequality, the Wirtinger inequality, and the spectrum of the Laplacian lead to the definition of a discrete spherical Laplacian enjoying spectral properties similar to its smooth counterpart.