A conditional limit theorem for high-dimensional ${\mathrm{\ell }}^p$ spheres

The study of high-dimensional distributions is of interest in probability theory, statistics and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ${\mathrm{\ell }}^p$ spaces and norms are of particular interest in this setting. In this paper, we establish a limit theorem for distributions on ${\mathrm{\ell }}^p$ spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ${\mathrm{\ell }}^p$ balls in a high-dimensional Euclidean space.