Finiteness principles for smooth convex functions

Let E subset of R-n be a compact set, and f:E -> R. How can we tell if there exists a convex extension F is an element of C-1,C-1(R-n) of f, i.e. satisfying F|(E)=f|(E)? Assuming such an extension exists, how small can one take the Lipschitz constant Lip(del F):=sup(x,y is an element of R)n,(|del F(x)-del F(y)|)(x not equal y)/(|x-y|)? We provide an answer to these questions for the class of strongly convex functions by proving that there exist constants k(#)is an element of N and C>0 depending only on the dimension n, such that if for every subset S subset of E, #S <= k(#), there exists an( eta)-strongly convex function F-S is an element of C-1,C-1(R-n) satisfying F-S|S=f|(S) and Lip(del F-S)<= M, then there exists an (eta)/(C)-strongly convex function F is an element of C-c(1,1)(R-n) satisfying F|(E)=f|(E), and Lip(del F)<= CM2/eta. Further, we prove a Finiteness Principle for the space of convex functions in C-1,C-1(R) and that the sharp finiteness constant for this space is k(#)=5.