Abstract. Given a function f : E → R with E ⊂ R n , we explain how to decide whether f extends to a C m function F on R n . If E is finite, then one can efficiently compute an F as above, whose C m norm has the least possible order of magnitude (joint work with B. Klartag).Let f : E → R be a function defined on a given (arbitrary) set E ⊂ R n , and let m ≥ 1 be a given integer. How can we decide whether f extends to a function The finite, effective versions of the above problems are basic questions about interpolation of data: Let E ⊂ R n be a finite set, and let f : E → R be given. Fix m ≥ 1. We want to find an interpolant, i.e., a function F ∈ C m (R n ) such that F = f on E. How small can we take the C m norm of an interpolant? How can we compute an interpolant whose C m norm is close to least possible? What if we require only that F and f agree approximately on E? What if we are allowed to discard a few points from E? An efficient solution to these problems would likely have practical applications. Joint work [19], [20] of B. Klartag and the author shows how to compute efficiently an interpolant F whose C m norm is within a factor C of least possible, where C is a constant depending only on m and n. Unfortunately, we do not know whether that constant C is absurdly large; we suspect that it is. To remedy this defect, and hopefully obtain results with practical applications, we pose the following sharper version of the interpolation problem.