Optimal Transport: A Semi-Discrete Approach

Some isoperimetric inequalities on RN with respect to weights |x|α

In this paper we consider the isoperimetric problem with double density in Euclidean space; that is, we study the minimisation of the perimeter among subsets of R n with fixed volume, where volume and perimeter are relative to two different densities. The case of a single density, or equivalently, when the two densities coincide, has been well studied in the last years; nonetheless, the problem with two different densities is an important generalisation, also in view of applications. We will prove the existence of isoperimetric sets in this context, extending the known results for the case of single density.

On the isoperimetric problem with double density

https://cronfa.swan.ac.uk/Record/cronfa51275/Download/0051275-13082019084641.pdf

The isoperimetric problem for a class of non-radial weights and applications

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

/pdf/isoperimetric-regions-in-rn-with-density-rp-25mtvwrz0d.pdf

Isoperimetric Regions in Rn with Density rp

Let I be a toric ideal. We say I is robust if its universal Grobner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.

Robust Graph Ideals

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 $r^p$

Let I be a toric ideal. We say I is robust if its universal Groebner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.

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

/pdf/the-convex-body-isoperimetric-conjecture-in-the-plane-3535x2oz7v.pdf

Bryan Brown

Papers

Isoperimetric Regions in Rn with Density rp

Robust Graph Ideals

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

Robust Graph Ideals

The Convex Body Isoperimetric Conjecture in the Plane