. According to the MathSciNet search engine, approximately 100 papers and books pertaining to the Brunn-Minkowski inequality have appeared since Das Gupta's 1980 article Brunn-Minkowski and its aftermath [56]. An attempt is made here to present an up-to-date survey. 1. Introduction About a century ago, not long after the first complete proof of the classical isoperimetric inequality was found,
... [Show full abstract] Minkowski proved the following inequality: V ((1 Gamma )K + L) 1=n (1 Gamma )V (K) 1=n + V (L) 1=n : (1) Here K and L are convex bodies (compact convex sets with nonempty interiors) in R n , 0 ! ! 1, V denotes volume, and + denotes vector or Minkowski sum. The inequality (1) had been proved for n = 3 earlier by Brunn, and now it is known as the Brunn-Minkowski inequality. It is a sharp inequality, equality holding if and only if K and L are homothetic. The Brunn-Minkowski inequality was inspired by issues around the isoperimetric problem, and was for a long time considered to be...