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Kirszbraun's theorem states that every Lipschitz map  S  → ℝ  n  , where  S ⊆ ℝ  m  , has an extension to a Lipschitz map ℝ  m   → ℝ  n   with the same Lipschitz constant. Its proof relies on Helly's theorem: every family of compact subsets of ℝ  n  , having the property that each of its subfamilies consisting of at most ( n  + 1) sets share a common point, has a non‐empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.

Definable versions of theorems by Kirszbraun and Helly