## Functional Isoperimetric Inequalities

In geometry the most fundamental inequalities are isoperimetric inequalities. In this post we will focus on dimension two, although all of the results extend to higher dimensions. In **R**^{2}, the perimeter and area of a region satisfy

(1) *P* ≥ (4π*A*)^{1/2},

with equality for a round disc. On the unit sphere, for 0 < *A* < 4π,

(2) *P* ≥ (*A*(4π-*A*))^{1/2},

with equality for a geodesic disc.

In analysis, the most fundamental inequalities relate functions and often their derivatives. Continue reading ‘Functional Isoperimetric Inequalities’ »