Gordon's inequality and condition numbers in conic optimization

摘要

The probabilistic analysis of condition numbers has traditionally beenapproached from different angles; one is based on Smale's program in complexitytheory and features integral geometry, while the other is motivated bygeometric functional analysis and makes use of the theory of Gaussianprocesses. In this note we explore connections between the two approaches inthe context of the biconic homogeneous feasiblity problem and the conditionnumbers motivated by conic optimization theory. Key tools in the analysis areSlepian's and Gordon's comparision inequalities for Gaussian processes,interpreted as monotonicity properties of moment functionals, and theirinterplay with ideas from conic integral geometry.