Frank Morgan

In Rⁿ or in a Riemannian manifold, one may consider regions R with density given by an integrable nonnegative function g, with volume ∫_R g. If everything is smooth, the perimeter is given by ∫_∂R g, or more generally by Stokes’ Theorem. For finite perimeter, these are the so-called normal currents of geometric measure theory. All of this can be done in a manifold with density f (unrelated to g).

In Rⁿ, if you allow regions with density, there is no isoperimetric optimum for given volume, because large balls with low (constant) densities approach perimeter 0; similarly in any space for which P/V has no minimum. In a space of finite volume, such as Gauss space Gⁿ (Rⁿ with Gaussian density) there is an optimum: the whole space with appropriate constant density has perimeter 0. At the other extreme, in R with density exp(x³), a left halfline with high density approaches perimeter 0.

Note that if a region with density is isoperimetric, it remains so for all multiples of that density.

In principle one may allow regions with variable density f, the variability contributing to the boundary, but since any such region is an integral of regions R_d = {f ≥ d} with constant density d, one need consider only regions with constant density.

For a manifold with density with isoperimetric profile P = I(V), the least-perimeter region with density with unit volume just minimizes cI(1/c); in a space of infinite volume its perimeter to volume ratio is the Cheeger constant inf P/V.

In R² with density exp(r²), where classical isoperimetric regions are balls about the origin, among regions with density the isoperimetric optimum for every area is the same disc of radius r ~ 1.1 (solution to exp(r²) = 2r² + 1) with appropriate constant density.

Proposition. In Rⁿ with smooth, uniformly log-convex radial density f(r), an isoperimetric region with density exists.

Proof. By Morgan-Pratelli (Thms. 3.3, 5.8), a classical, compact  region R of least perimeter P exists for every given volume V. By Kolesnikov-Zhdanov [ZD] (see also Howe Cor. 3.7) and Figalli-Maggi [FM], for large and small volumes R is a ball about the origin. Consequently, as V approaches 0 or infinity, P/V goes to infinity. (For small balls this is trivial. For large balls it follows from the fact that dP/dV = (1/P)dP/dr ≥ (1/f)df/dr = d(log f)/dr goes to infinity.) Since P and hence P/V are continuous functions of V, P/V attains a minimum, providing an isoperimetric region with density.

[FM]  A. Figalli and F. Maggi, On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations, to appear.

[KD] Alexander V. Kolesnikov and Roman I. Zhdanov, On isoperimetric sets of radially symmetric measures, Concentration, functional inequalities and isoperimetry, 123–154, Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011; arXiv (2010), Cor. 6.8.

Remark. In R with density exp(x), for given volume, left halflines are classically isoperimetric and halflines with constant density all have the same perimeter, as do their integral averages, yielding lots of isoperimetric optima.