Reinhardt polygons: https://en.wikipedia.org/wiki/Reinhardt_polygon
These polygons have equal side lengths and are inscribed in Reuleaux polygons. Among all convex polygons with the same number of sides (any number that is not a power of two), they have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.
New Wikipedia article.
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