Marjorie Drake
MIT
Title
Convexity in Whitney Problems
Abstract
Let $E \subset \mathbb{R}^n$ be a compact set, and $f:E \to \mathbb{R}$. How can we tell if there exists a smooth convex extension $F \in C^{1,1}(\mathbb{R}^n)$ of $f$, i.e. satisfying $F|_E = f|_E$? Assuming such an extension exists, how small can one take the Lipschitz constant $\text{Lip}(\nabla F): = \sup_{x,y \in \mathbb{R}^n, x \neq y} \frac{|\nabla F(x) - \nabla F(y)|}{|x-y|}$? I will provide an answer to these questions for the non-linear space of strongly convex functions by presenting recent work of mine proving there is a Finiteness Principle for strongly convex functions in $C^{1,1}(\mathbb{R}^n)$. This work is the first attempt to understand the constrained interpolation problem for \emph{convex} functions in $C^{1,1}(\mathbb{R}^n)$, building on techniques developed by P. Shvartsman, C. Fefferman, A. Israel, and K. Luli to understand whether a function has a smooth extension despite obstacles to their direct application. We will finish with a discussion of challenges in adapting my proof of a Finiteness Principle for the space of convex functions in $C^{1,1}(\mathbb{R})$ ($n=1$) to higher dimensions.
