Finiteness Principles for Smooth Convex Functions

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 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 $\text{Lip}(\nabla F): = \sup_{x,y \in \mathbb{R}^n, x \neq y} \frac{|\nabla F(x) - \nabla F(y)|}{|x-y|}$? We provide an answer to these questions for the class of strongly convex functions by proving that there exist constants $k^\# \in \mathbb{N}$ and $C>0$ depending only on the dimension $n$, such that if for every subset $S \subset E$, $\#S \leq k^\#$, there exists an $η$-strongly convex function $F^S \in C^{1,1}(\mathbb{R}^n)$ satisfying $F^S|_S=f|_S$ and $\text{Lip}(\nabla F^S) \leq M$, then there exists an ${\fracη{C}}$-strongly convex function $F \in C^{1,1}_c(\mathbb{R}^n)$ satisfying $F|_E = f|_E$, and $\text{Lip}(\nabla F) \leq C M^2/η$. Further, we prove a Finiteness Principle for the space of convex functions in $C^{1,1}(\mathbb{R})$ and that the sharp finiteness constant for this space is $k^\#=5$.