# Continuous extension preserving modulus of continuity

Let $$X$$ be a (non-empty) compact subset of $$D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$$, and let $$f:X\rightarrow Y$$ be uniformly continuous; for some metric space $$Y$$. Are there any known result guaranteeing that $$f$$ can be extended to a uniformly continuous function $$F:D(0,M)\rightarrow Y$$ such that the modulus of continuity of $$f$$ is (point-wise) greater-than or equal to the modulus of continuity of $$F$$?

• If $X$ is the sphere, then possibility of extension means that $n-1$-th homotopy group of $Y$ vanish. Also, it doesn't say in the body of the question that modulus of continuity has to be preserved (as you mentioned in the title). Could you please clarify?
– erz
Mar 15, 2021 at 6:12
• I added the requested clarification. However, I don't see the homotopic argument. Mar 15, 2021 at 10:23
• one of the definition of having vanishing $n-1$-th homotopy group is precisely existence of extension of any continuous map $f:S^{n-1}\to Y$ to $F:D_{n}\to Y$
– erz
Mar 15, 2021 at 16:47
• See for example Hatcher - Algebraic topology, chapter on homotopy groups
– erz
Mar 16, 2021 at 17:42
• I did not claim it does. It's just a necessary condition
– erz
Mar 19, 2021 at 18:11