A relative isoperimetric inequality for certain warped product spaces

TL;DR

This paper establishes a new isoperimetric inequality in warped product spaces with convex warping functions, providing solutions to a Dido problem and volume bounds for graphs over vertical fibers.

Contribution

It proves a relative isoperimetric inequality for regions in warped product spaces with convex warping functions, extending classical results to this setting.

Findings

Proves a lower bound for the volume of graphs over vertical fibers.

Solves a Dido problem for graphs in warped product spaces.

Shows volume bounds depend on the warping function's behavior.

Abstract

Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a $k$--dimensional region $F \subset \{b\} \times N$, and the horizontal graph $C \subset \mathbb{R} \times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph over $F$ that traps the same $(1+k)$--volume above $F$ as $C$. We use this to solve a Dido problem for graphs over vertical fibers, and show that, if the warping function is unbounded on the set of horizontal values above a vertical fiber, the volume trapped above that fiber by a graph $C$ is no greater than the $k$--volume of $C$ times a constant that depends only on the warping function.