Finiteness Principles for Smooth Convex Functions

Let E ⊂ Rⁿ be a compact set, and f:E → R. How can we tell if there exists a convex extension F ∈ C1,1(Rⁿ) of f, i.e. satisfying F|E = f|E? Assuming such an extension exists, how small can one take the Lipschitz constant Lip(∇ F): = x,y ∈ Rⁿ, x ≠ y |∇ F(x) - ∇ 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^# ∈ N and $C>0$ depending only on the dimension n, such that if for every subset S ⊂ E, #S ≤ k^#, there exists an η-strongly convex function FS ∈ C1,1(Rⁿ) satisfying FS|S=f|S and Lip(∇ FS) ≤ M, then there exists an η/C-strongly convex function F ∈ C1,1c(Rⁿ) satisfying F|E = f|E, and Lip(∇ F) ≤ C M²/η. Further, we prove a Finiteness Principle for the space of convex functions in C1,1(R) and that the sharp finiteness constant for this space is k^#=5.