Archive for the ‘Math’ Category.

Indian Science Camp

Bangalore12I just had a great time at the amazing conference for 800 top Indian high school science students. Here I provide some contacts and information for my young new Indian friends and all; some pictures, including some wonderful pentagonal tilings I found on the path to the Guest House, the ever busy and calm organizer Kaushal Verma, and my student host Devang Rammohan, who met me at the airport at 5 am and took me back at midnight. I’m grateful to all the organizers and participants for their role in this inspiring vision of the future of science in India and beyond. Continue reading ‘Indian Science Camp’ »

Sharp Isoperimetric Bounds for Convex Bodies

Emanuel Milman [M2, 2013] provides very general sharp lower bounds on perimeter to enclose prescribed volume, including the following special case of convex bodies in Rn. Here we give an alternative proof for that special case as suggested by Milman [M2, §7.2]. For sharp upper bounds see my post [Mo1] on the Convex Body Isoperimetric Conjecture. Continue reading ‘Sharp Isoperimetric Bounds for Convex Bodies’ »

Optimal Transportation with Constraint

In Barcelona, Robert McCann talked about his work with Jonathan Korman and Christian Seis on optimal transportation with a constraint h(x,y) on the flow from x to y. A constant constraint h means that an x must be spread out over at least fraction 1/h of the target; there is not the capacity to send it all to the most desirable spot. Here we present a simplified extension of some of their results.  Continue reading ‘Optimal Transportation with Constraint’ »

Geometry and PDEs in Barcelona

Sixty mathematicians and students gathered in Barcelona at the Center for Research in Mathematics CRM for a Conference on Qualitative and Geometric Aspects of Elliptic PDEs, proficiently organized by Xavier Cabré, Daniele Castorina, Manel Sanchón, and Enrico Valdinoci.

Barcelona01

In my talk I mentioned a new isoperimetric theorem by Xavier Cabré and his students Xavier Ros-Otón and Joaquim Serra, which they describe in this video. Continue reading ‘Geometry and PDEs in Barcelona’ »

Pradham ’13 on Wallet Paradox

The famous Wallet Paradox invites two similar individuals to lay their wallets on the table, the one with the lesser amount of money to win both. Paradoxically, each might reason: “I have the advantage, because if I lose, I lose just what I have, but if I win, I win more than I have.” A follow-up analysis assumes that each has the same expected amount of money and asks for the best probability distribution or “best strategy” with that given mean. The following note is based on a senior colloquium talk. Continue reading ‘Pradham ’13 on Wallet Paradox’ »

Isoperimetric Profile Continuous?

Given a smooth Riemannian manifold, the isoperimetric profile I(V) gives the infimum perimeter of smooth regions of volume V.

Proposition 1. In a compact smooth Riemannian manifold of dimension at least two, the isoperimetric profile is continuous. Continue reading ‘Isoperimetric Profile Continuous?’ »

Distance to Boundary of Manifold with Density

Jian Ge’s recent ArXiv post on “Comparison theorems for manifolds with mean convex boundary,” Theorem 0.1, has a generalization to manifolds with density, here within a factor of 2 of sharp for constant density: Continue reading ‘Distance to Boundary of Manifold with Density’ »

Isoperimetric Regions with Density

In Rn or in a Riemannian manifold, one may consider regions R with density given by an integrable nonnegative function g, with volume ∫g. If everything is smooth, the perimeter is given by ∫∂R g, or more generally by Stokes’ Theorem. For finite perimeter, these are the so-called normal currents of geometric measure theory. All of this can be done in a manifold with density f (unrelated to g).

In Rn, if you allow regions with density, there is no isoperimetric optimum for given volume, because large balls with low (constant) densities approach perimeter 0; similarly in any space for which P/V has no minimum. In a space of finite volume, such as Gauss space Gn (Rn with Gaussian density) there is an optimum: the whole space with appropriate constant density has perimeter 0. At the other extreme, in R with density exp(x3), a left halfline with high density approaches perimeter 0.

Note that if a region with density is isoperimetric, it remains so for all multiples of that density.

In principle one may allow regions with variable density f, the variability contributing to the boundary, but since any such region is an integral of regions Rd = {fd} with constant density d, one need consider only regions with constant density.

For a manifold with density with isoperimetric profile P = I(V), the least-perimeter region with density with unit volume just minimizes cI(1/c); in a space of infinite volume its perimeter to volume ratio is the Cheeger constant inf P/V.

In R2 with density exp(r2), where classical isoperimetric regions are balls about the origin, among regions with density the isoperimetric optimum for every area is the same disc of radius r ~ 1.1 (solution to exp(r2) = 2r2 + 1) with appropriate constant density.

Proposition. In Rn with smooth, uniformly log-convex radial density f(r), an isoperimetric region with density exists.

Continue reading ‘Isoperimetric Regions with Density’ »

Isoperimetric Regions Bounded

An ingredient in proving the existence of isoperimetric regions and clusters of prescribed volume(s) is the boundedness of isoperimetric regions of smaller volume(s). One proof of boundedness is by monotonicity. The proof in my Geometric Measure Theory book (Lemma 13.6) uses a non-sharp isoperimetric inequality for small volume and has the advantage of applying to convex integrands more general than area; the requisite isoperimetric inequality follows immediately from the isoperimetric inequality for area. All of this works equally well in the presence of a density. For more delicate results see

Frank Morgan and Aldo Pratelli, Existence of isoperimetric regions in Rwith density, Ann. Global Anal. Geom. (2012); arXiv.org (2011)

and Cinti and Pratelli.

My book comments that existence similarly holds in any smooth Riemannian manifold M with compact quotient M/G by a group of isometries. Monotonicity still yields boundedness; even with density (bounded above and below by compactness) the classical mean curvature is bounded and hence classical monotonicity applies. To use my alternate proof, which applies to more general integrands, one needs the isoperimetric inequality for small volume, which follows immediately from such an isoperimetric inequality for area without density. For the latter in a more general setting, see Theorem 2.1 of Morgan and Ritoré “Isoperimetric regions in cones“ after Berard and Meyer. In this simpler setting of M/G compact, you can just cover M with congruent balls with a bounded number of balls meeting each point and apply the relative isoperimetric inequality in each ball. Incidentally the same covering can be used to replace the division into cubes in the existence proof 13.7 in my book.

New Journal “Analysis and Geometry in Metric Spaces”

Analysis and Geometry in Metric Spaces” is one of a number of new open access journals to be funded by author fees, such as the Gowers-Tao Forum of Mathematics. The idea is that long-term it will be much cheaper for institutions to pay author fees than subscription fees, but the transition will be difficult, since while the major journals are on the old model, institutions cannot drop their subscriptions and may not pay author fees, and authors would rather publish for free in established journals.

“Analysis and Geometry in Metric Spaces” provides a journal for an important and rapidly growing modern area of mathematics and to its credit it does it on this new open access model of the future. I wish my good friend and the editor Manuel Ritoré of the extraordinary Department of Geometry and Topology at the University of Granada every success.