12.  Variational Geometry

The nineteenth general mathematical framework is that of variational geometry, which is a branch of mathematics combining variational calculus and geometry. The calculus of variations (or variational calculus) is a branch of mathematical analysis that uses variations (variations of functionals)—which are “small” changes in the values of functionals—to find their maxima and/or minima.

The typical problem is to find a subset that is “optimal” in the sense that it minimizes a set-theoretic functional F while satisfying certain constraints (G(X) = ct), e.g., the following variational problem:

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