Sard's Theorem for Min-functions According to Yomdin

Consider a 𝐶𝑘 function 𝐹:𝑀𝑚×𝑁𝑛⟶ℝ defined on the product of the manifold 𝑀𝑚

of dimension 𝑚 with the compact manifold 𝑁𝑛 of dimension 𝑛.

The function defined by inf𝑁𝑛𝐹:𝑀𝑚⟶ℝ is locally Lipschitz. Yomdin proved (in 1984) that if 𝑘𝑘(𝑚,𝑛), where the integer 𝑘(𝑚,𝑛) depends only on 𝑚 and 𝑛, then for almost every 𝑟𝑅, the preimage {𝑥 | inf𝑁𝑛𝐹(𝑥)=𝑟} is a codimension one submanifold of 𝑀𝑚.

Yomdin's proof relies on his approach to almost critical values, yielding quantitative versions of Sard's theorem. This bound has subsequently been improved by several authors, always using Yomdin's methods.

We shall give a direct proof based on the classical Morse--Sard theorem.