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Geometric Inequalities

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Abstract

The classical isoperimetric inequality states that among all (convex) sets with a given perimeter, the circular disk has the greatest area. This is a very old result that has been generalized in many directions. Apart from the fact that one does not need convexity here, essentially the same result holds in all scenarios in which the notions of “perimeter” and “area” can be naturally defined. The embedding space can also be varied: similar inequalities are true in higher dimensions (balls have maximum volume among all bodies with a given surface area), and problems of this type have also been considered in many spaces other than the Euclidean. Moreover, many analogous inequalities have been established concerning other measures besides the perimeter and the area. For the extensive literature on the many aspects of these problems see [ScA00], [BuZ88], [Tal93], [Lu93], [Fl93], [Ha57]. In this section, we concentrate on isoperimetric problems in discrete geometry, and avoid most questions that largely belong to convexity, differential geometry, or geometric measure theory.

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(2005). Geometric Inequalities. In: Research Problems in Discrete Geometry. Springer, New York, NY. https://doi.org/10.1007/0-387-29929-7_12

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