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The voronoi flower of a compact set $F\subseteq\mathbb{R}^d$ with respect to a point $x$ is given by $$\mathcal{F}_x(F):=\bigcup_{y\in F}B_{\|y-x\|}(y).$$

I need to show that $\mathcal{F}_x(F)=\mathcal{F}_x(\text{conv}(F))$. This is a result that is generally known, but I haven't been able to find a proof of it.

So far, I've tried to prove it directly. For $F=\{a,b\}$ and $y=\alpha a+(1-\alpha)b$. Assuming $\|y-z\|<\|y-x\|$ and $\|a-z\|>\|a-x\|$ to show that $\|b-z\|<\|b-x\|$, but when I try to apply the triangle inequality I can't get there.

venom
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Mar 08 '23 at 10:32
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    This property is more or less established on page 716 of this document https://ahl.centre-mersenne.org/item/10.5802/ahl.86.pdf – Jean Marie Mar 17 '23 at 22:27

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