Frank Morgan

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 R², the perimeter and area of a region satisfy

(1)            P ≥ (4πA)^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’ »