Diameter
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
The largest polygon of diameter one
Among pentagons whose points are never more than one unit apart, the regular pentagon has the most area, and the same is true for every odd number of sides. For six sides it is false: a lopsided hexagon found in 1975 beats the regular one by four per cent. Symmetry wins half the cases and loses the other half, and the reason is parity.
The longest fence of diameter one
Among polygons with n sides and diameter one, the one with the most area is regular only when n is odd. Ask for the most perimeter instead and the answer is kinder: Karl Reinhardt proved in 1922 that the longest perimeter is 2n sin(π/2n) whenever n has any odd factor at all, reached by polygons cut from curves of constant width, many of them not regular. Only when n is a power of two does every polygon fall short — and the best quadrilateral, found by searching every one, is an equilateral triangle with one corner bent out.
Named alongside it
The objects these essays reach for when they reach for this one.
Constant widthExtremal problemIsodiametric inequalityIsoperimetric problemLocal searchOptimisationPerimeterRegular polygonReuleaux triangleSymmetry