Archive for July 2013

Isoperimetric Profile Continuous?

Note added 5 June 2016. A noncompact counterexample is given by Nardulli and Pansu, arxiv.org. On the positive side, see the comment below by Milman and Flores/Nardulli [FN].

Note added 21 March 2018. A 2D (noncompact) counterexample is given by Papasoglu and Swenson [PS], via expander graphs.

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?’ »

Alaskan Bridge Cruise with Billy Miller

CruiseBlog1 My mom and I just had a great time on the first ACBL Regional at Sea, an Alaskan cruise with the inimical Billy Miller. My mom won enough gold points to become a Life Master. In that fateful session Thursday afternoon, July 18, 2013, as West she played one hand at 3N which should only make 2N and she somehow made 5N:

Screen shot 2013-02-26 at 5.14.10 PM

Continue reading ‘Alaskan Bridge Cruise with Billy Miller’ »

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’ »