Here is a proof of Kirszbraun theorem for a case where E is bounded and convex :
If we assume that E is a bounded convex set of R^n, then the proof of Kirszbraun's theorem becomes easy:
We start by setting K = closure of E, which is a compact convex set.
We extend f to K by uniformly continuous extension: for every point x ∈ K, we take a sequence (xn)n of points in E converging to x, and we define f°(x) = lim (n→∞) f(xn), demonstrating that the limit exists and does not depend on the sequence The function f° thus constructed is indeed L-Lipschitz.
Finally, we introduce the orthogonal projection PK onto K and set F(x) := f°(PK(x)), ∀x ∈ R d . It is clear that F extends f, and furthermore, since PK is 1-Lipschitz, we see that F is L-Lipschitz."
-but i still have some unclear points about the proof:
- Where exactly did we utilize the assumption that E is bounded and convex?
- How do we prove the assertion that "the limit exists and does not depend on the sequence"?
- Can you provide further clarification on the final step of the proof?