Brunn–Minkowski theory

Summary

Brunn–Minkowski theory is the part of convex geometry that studies convex bodies through the interplay of volume and Minkowski addition. Its central objects are the mixed volumes, the coefficients of the polynomial that gives the volume of a Minkowski combination of convex bodies. Its central results are inequalities between them, starting from the Brunn–Minkowski theorem and leading to the Alexandrov–Fenchel inequality. Surface area, mean width, quermassintegrals and the isoperimetric inequality all fit into this framework.