I will discuss a circle of questions about the geometry of sums of large sets in R^n endowed with the Gaussian measure, related to the convexity problem of Talagrand: if a subset A of R^n has Gaussian measure at least 2/3 say, for which integer q does A+...+A (q times) contain a large convex body? I will explain why it is closely connected to the problem of understanding sums of non-independent Gaussian random vectors. I will discuss some sharp results and partial answers.