Optimal Transport: A Semi-Discrete Approach

Abstract We prove the validity of the ε - ε β {\\varepsilon-\\varepsilon^{\\beta}} property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets

We prove the validity of the $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.

The $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.

Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\in\Omega$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$

On the isoperimetric problem with perimeter density r^p

We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.

Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

The Convex Body Isoperimetric Conjecture states that the least perimeter needed to enclose a volume within a ball is greater than the least perimeter needed to enclose the same volume within any other convex body of the same volume in Rn. We focus on the conjecture in the plane and prove a new sharp lower bound for the isoperimetric profile of the disk in this case. We prove the conjecture in the case of regular polygons and show that in a general planar convex body, the conjecture holds for small areas. Acknowledgements: This paper is the work of the Williams College NSF “SMALL” 2013, 2014, and 2015 Geometry Groups. We thank our advisor Professor Frank Morgan for his support. We would like to thank the National Science Foundation, Williams College, and the MAA for supporting the “SMALL” REU and our travels to MathFest 2013, MathFest 2014, and MathFest 2015. Page 10 RHIT Undergrad. Math. J., Vol. 18, No. 2

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The Convex Body Isoperimetric Conjecture in the Plane

We consider optimal transportation with constraint, as did Korman and McCann (2013, 2015), provide simplifications and generalizations of their examples and results, and provide some new examples and results.

https://msp.org/involve/2019/12-1/involve-v12-n1-p01-s.pdf

Optimal transportation with constant constraint

https://msp.org/involve/2018/11-2/involve-v11-n2-p03-s.pdf

Alyssa Loving

Papers

Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$

The Convex Body Isoperimetric Conjecture in the Plane

Optimal transportation with constant constraint

Double bubbles in hyperbolic surfaces