# Socially Distant Polygons

Imagine a polygon, like the blue one below. Where should you stand in that polygon such you are as far as possible from your nearest vertex? This seems like a useful thing to be able to calculate in 2020 (hope this blog post ages well). Ostensibly this should be easy to compute too, but it’s not obviously that the red dot in the image below is in fact the furthest point from any of the polygon vertices.

A first guess might be to stand half the longest edge length away from one of the vertices. But unfortunately that doesn’t work. If you try a square you’ll see it’s best to stand in the centre which is slightly further away from each corner. This shows that we need to consider points on the interior of the polygon to make sure they are covered too. Stated another way we are trying to calculate the minimum radius circle `R`

, such that the area of the union of circles positioned at `(xi,`

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