Speaker:Prof. Béla Bollobás (DPMMS)
Venue: Old Combination Room, Trinity College
Time: 15/10/2007 20:30, drinks from 20:15

The sum S of k sets of integers A₁,A₂,…,A_k is defined as S=A₁+A₂+…+A_k={a₁+…+a_k:a_i&isin A_i for every i}. For non-empty finite sets A_i, it is easily seen that the size |S| of the sum is at least |A₁|+…+|A_k|-k+1. There are similar classical inequalities for subsets of additive groups – indeed, such an equality was proved by Cauchy and rediscovered by Davenport. In the talk, aimed at first year undergraduates, we shall consider some related inequalities concerning the minima and maxima of set sums. For example, given |A₁+A₂|, |A₁+A₃| and |A₂+A₃|, what can we say about the sum of the three sets? As we shall see, these problems are intimately connected to inequalities involving the volumes of “canonical” projections of a body in R^k.