Geometric Analysis and Topology Seminar

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Approximation of convex functions

Speaker: Piotr Hajlasz, University of Pittsburgh

Location: Warren Weaver Hall 512

Date: Friday, May 17, 2024, 11 a.m.

Synopsis:

It has been known for at least thirty years, that convex functions have the {\em $C^2$-Lusin property}, meaning that if $u:\mathbb{R}^n\to\mathbb{R}$ is convex, then for every $\varepsilon>0$, there is a function $v\in C^2(\mathbb{R}^n)$ such that $|\{u\neq v\}|<\varepsilon$. However, in general, $v$ cannot be convex (there are counterexmples). The problem of approximating a convex function by $C^2$-convex functions in the Lusin sense has remained unresolved since the nineties.

In the talk I will discuss recent results about approximation of convex functions by $C^{1,1}_{\rm loc}$ convex functions and approximation of locally strongly convex functions by $C^2$-locally strongly convex functions, both uniformly and in the Lusin sense. 

As a byproduct of the methods, I will show a new and very simple proof of the Alexandrov theorem about second order differentiability of convex functions.