Polynomial-Time Rademacher Theorem, Porosity and Randomness

The main result of this paper is a polynomial time version of Rademacher\u27s theorem. We show that if z is p-random, then every polynomial time computable Lipschitz function f:R^n->R is differentiable at z. This is a generalization of the main result of [Nies, STACS2014]. To prove our main result, we introduce and study a new notion, p-porosity, and prove several results of independent interest. In particular, we characterize p-porosity in terms of polynomial time computable martingales and we show that p-randomness in R^n is invariant under polynomial time computable linear isometries