Randomness and Differentiability of Convex Functions

Abstract

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies differentiability of computable Lipschitz functions of several variables. Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.

Polynomial Space Randomness in Analysis

“…Recently, research in algorithmic randomness has used computable analysis to study the connection between randomness and classical analysis [1,5,6,7,14,15,20]. With the rise of measure theory, many fundamental theorems of analysis have been "almost everywhere" results.…”