We discuss two problems in stochastic geometry:

1. Sylvester's problem is a classical question that asks for the probability that d+2 random points in R^d are in convex position, that is, that none of them is in the convex hull of the others. We compute this probability when the points are the steps of a random walk. Remarkably, the probability depends only on dimension and is independent of the increment distribution of the walk, provided that it satisfies a mild nondegeneracy condition.

2. An exact formula for the mean volume of the convex hull of d-dimensional Brownian motion at a given time has been known since 2014. What can be said about the inverse volume process? In other words, how much time, on average, is required for the convex hull to attain a given volume? We establish two-sided bounds that capture the correct order of asymptotic growth in the dimension.