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2016•arXiv (Cornell University)Open access

The Log Convex Density Conjecture in Hyperbolic Space

Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu

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Abstract

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We generalize his results from $\mathbb{R}^n$ to $\mathbb{H}^n$ with related but different volume and perimeter densities.

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The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We generalize his results from $\mathbb{R}^n$ to $\mathbb{H}^n$ with related but different volume and perimeter densities.

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Available abstract

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We generalize his results from $\mathbb{R}^n$ to $\mathbb{H}^n$ with related but different volume and perimeter densities.

Key concepts: Isoperimetric inequality, Perimeter, Conjecture, Regular polygon, Mathematics, Combinatorics, Convex body, Space (punctuation)

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