I will discuss a circle of questions about sums of large sets in R^n endowed with the Gaussian measure, related to the convexity conjecture of Talagrand. If a subset A of R^n has Gaussian measure at least 2/3 say, how much larger is A+...+A (q times)? I will explain how this question is connected to the problem of understanding sums of Gaussian random vectors, and I will give some partial answers. I might also mention known connections to analogous combinatorics problems.