A relative isoperimetric inequality for certain warped product spaces
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Shawn Rafalski
Abstract
Given a warped product space ℝ ×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 ⊂ {b} × N , and the horizontal graph C ⊂ ℝ ×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.
© 2012 by Walter de Gruyter GmbH & Co.
Articles in the same Issue
- Masthead
- Barycenters in Alexandrov spaces of curvature bounded below
- Isoperimetric problems in sectors with density
- Around A. D. Alexandrov’s uniqueness theorem for convex polytopes
- A relative isoperimetric inequality for certain warped product spaces
- Markov’s inequality in the o-minimal structure of convergent generalized power series
- Minimal area conics in the elliptic plane
- Geometry of semi-tube domains in ℂ2
- Flag transitive c:F4(2)-geometries
- Majorana representations of L3(2)
- A characterization of multiple (n – k)-blocking sets in projective spaces of square order
Articles in the same Issue
- Masthead
- Barycenters in Alexandrov spaces of curvature bounded below
- Isoperimetric problems in sectors with density
- Around A. D. Alexandrov’s uniqueness theorem for convex polytopes
- A relative isoperimetric inequality for certain warped product spaces
- Markov’s inequality in the o-minimal structure of convergent generalized power series
- Minimal area conics in the elliptic plane
- Geometry of semi-tube domains in ℂ2
- Flag transitive c:F4(2)-geometries
- Majorana representations of L3(2)
- A characterization of multiple (n – k)-blocking sets in projective spaces of square order
