Let $K, K'$ be two compact convex subsets of $\mathbb{R}^n$ such that $K' \subset \mathrm{int}(K)$, and $f,g: K \to \mathbb{R}^n$ be two continuous maps.
I would like to find:
A continuous map $\phi: K \to \mathbb{R}^n$ such that $\phi|_{K'} = f|_{K'}$, and such that $\phi|_{\partial K} = g|_{\partial K}$.
A continuous map $H: [0,1] \times K \to \mathbb{R}^n$ such that for every $x \in K$, $H(0,x) = f(x)$ and $H(1,x) = \phi(x)$ (a homotopy between $f$ and $h$), and such that for every $t \in [0,1]$, $H(t,\cdot)|_{K'} = f|_{K'} (= \phi|_{K'})$.
For the second point, I think that the standard homotopy $H: [0,1] \times K \to \mathbb{R}^n, (t,x) \mapsto (1-t)f(x)+t\phi(x)$ works, right ? However, for the first point, I don't know at all how to construct the map $\phi$...
Do you have any clue ? Thank you !