Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, for instance in the compressed sensing theory. In this talk we will show that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution and lead to a phase transition. Our approach is based on a variety of techniques, including: (i) Steiner formulae for closed convex cones, (ii) Stein's method and second order Poincaré inequality, and (iii) concentration of measure estimates.
This work is joint with Laryy Goldstein (Southern California) and Giovanni Peccati (Luxembourg).