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Consider two compact convex sets $C_1, C_2 \subset \mathbb{R}^n$ such that $C_2 \subset C_1$. Let us denote by $\partial C_1$ and $\partial C_2$ their boundaries, that satisfy and $\partial C_1 \cap \partial C_2 = \varnothing$.

Consider two continuous, bounded, functions $f_1: C_1 \rightarrow \mathbb{R}^n$ and $f_2: C_1 \rightarrow \mathbb{R}^n$.

Consider a continuous, bounded, function $f: C_1 \rightarrow \mathbb{R}^n$ such that: $f(y) = f_1(y) \ \ \forall y \in \partial C_1$

$f(y) = f_2(y) \ \ \forall y \in \partial C_2$

1) Prove that there exists a continuous function $g: C_1 \rightarrow \mathbb{R}_{\geq 0}$ such that:

$ g(y) = 0 \ \ \forall y \in \partial C_1 $

$ g(y) = 1 \ \ \forall y \in \partial C_2 $

$ f(x) = ( 1-g(x) ) f_1(x) + g(x) f_2(x) \ \ \forall x \in \text{closure}(C_1 \setminus C_2) $

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    Fixed. In view of the problem statement, there is no substantial reason for not having $f_2$ defined on the whole $C_1$.2012-11-14

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Here is a hint. Let $\rho = d(\partial C_1, \partial C_2) \equiv \min \{\|x_1-x_2\|_2: x_1 \in \partial C_1, x_2\in \partial C_2\}$. Define $h(y) = d(y, C_2) = \min \{\|x-y\|_2: x\in \partial C_2\}$. Now let $g(y) = h(y)/\rho$ if $h(y) \leq \rho$, and $g(y) = 1$, otherwise.

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    Moreover, why the third property we are seeking ($f = (1-g)f_1 + g f_2$) should be necessarily satisfied with your particular $g$?2012-11-15