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We prove that the trace of the space C1,ω(Rn) to an arbitrary closed subset X ⊂ Rn is characterized by the following “finiteness” property. A function f : X → R belongs to the trace space if and only if the restriction f |Y to an arbitrary subset Y ⊂ X consisting of at most 3·2n−1 can be extended to a function fY ∈ C1,ω(Rn) such that sup{‖fY ‖C1,ω : Y ⊂ X, cardY ≤ 3 · 2n−1} <∞. The constant 3 · 2n−1 is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right. 1. Main results The results of the paper are concerned with the following problem having its origin in two classical papers of Hassler Whitney [W1], [W2] which appeared in 1934. Let C(R) be the space of k-times continuously differentiable functions f satisfying

Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions