Last time around we saw some interesting and non-intuitive behaviour of volumes and surfaces in high dimensions. Keeping in line with this, we will discuss the Brun-Minkowski theorem, and use it to prove similar concentration results in various settings: surface areas on the n-dimensional ball, the Gaussian Measure in R^n, and the Hamming Cube C_n. If time permits, I will mention a result on metric embeddings that relies on such a concentration of measure phenomenon.
