Let L m,p (R n ) be the Sobolev space of functions with m th derivatives lying in L p (R n ). Assume that n < p < ∞. For E ⊂ R n , let L m,p (E) denote the space of restrictions to E of functions in L m,p (R n ). We show that there exists a bounded linear map T : L m,p (E) → L m,p (R n ) such that, for any f ∈ L m,p (E), we have Tf = f on E. We also give

Let E ⊂ R n be a compact set, and f : E → R. How can we tell if there exists a convex extensionAssuming such an extension exists, how small can one take the Lipschitz constant Lip(∇F ) := sup x,y∈R n ,x =yWe 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 functionand Lip(∇F ) ≤ CM 2 /η. Further, we prove a Finiteness Principle for the space of convex functions in C 1,1 (R) and that the sharp finiteness constant for this space is k # = 5.

We present a coordinate-free version of Fefferman's solution of Whitney's extension problem in the space C m−1,1 (R n ). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. In a follow-up paper, we apply these ideas to study extension problems for a class of sub-Riemannian manifolds where global coordinates may be unavailable.Recall that B∈W θ B = 1 on B and x 0 ∈ B. Thus, B∈W J x 0 θ B = J x 0 (1) = 1. Therefore, (C) J x 0 F = P . By a standard technique we extend the function F ∈ C m−1,1 ( B) to a function in C m−1,1 (R n ) with norm bounded by C F C m−1,1 ( B) ≤ C ′ M -by abuse of notation,

We prove a finiteness principle for interpolation of data by nonnegative C^m and C^{m−1,1} functions. Our result raises the hope that one can start to understand constrained interpolation problems in which, e.g., the interpolating function F is required to be nonnegative.