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If $C$ is a convex compact set in $\mathbb{R}^n$, we know that we can define the projection on $C$, $p : \mathbb{R}^3 \setminus C \to C $, such that :

\begin{equation} \text{d}(x, p(x)) = \min_{y \in C} \text{d}(x,y) \end{equation}

Now is is true that for $x_1, x_2 \in C$, if $\text{d}(x, x_1) \le \text{d}(x, x_2)$, then :

\begin{equation} \text{d}(p(x), x_1) \le \text{d}(p(x), x_2) \end{equation}?

It seems to be true for every sketch I make, but I can't prove it, nor find a counterexample.

Plutoro
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hHhh
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1 Answers1

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Counterexample based on a plane picture but that can be easily extended to dimension $n$. The distance is the standard euclidean one.

enter image description here

mathcounterexamples.net
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