Let $K\subset\mathcal{R}^n$ be nonempty, convex and compact. Define the distance from a point to $K$ as $d(x,K)=\inf_{y\in K}d(x,y)$. Let $W=\{x\in \mathcal{R}^n\colon d(x,K)<r\}$. By drawing little diagrams in $\mathcal{R}^2$ my intuition tells me $W$ should be convex. Is it true? If so, how can I prove it rigorously? If not, could you please provide a counter example?
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