Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let (Mi,pi) be a sequence of n-dimensional Riemannian manifolds with lower Ricci curvature bound −1. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there exists a p∈X and subsequence of (Mi,pi) converging to (X,p) in the pointed Gromov-Hausdorff sense? Answer AttributionSource : Link , Question Author : dg.jan , … Read more

Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non overlapping hyperspheres whose centers are the closest to $p$ under the Euclidean metric (although I’m eventually interested in exploring other metrics as … Read more

Is positively curved Alexandrov surface isometrically embeddable in R3\mathbb R^3?

I guess it is not. The example I have in mind is: X2 is the spherical suspension of a circle S1(t) of length 0<t<2π. Then X has constant curvature =1 except at two suspension points, say N and S. But I cannot convince myself, since it seems this manifold can be approximated by a sequence … Read more

Name for metric spaces with useful unique-ball-intersection property?

When dealing with the problem of extending a Lipschitz function f:A→Y between metric spaces across an inclusion A↪X, one often imposes (conditions which imply) the following property on the target space Y. I’d like to know if this property has a name. Let me describe our property: there exists a uniform bound d∈(0,∞) so that … Read more

Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of the original dot. The countable sets of such dots have some property, let call it “class”. … Read more

Optimal instructions for the modular construction of rectlinear Lego structures

Let X be a compact (or periodic) union of integer translates of unit cubes such that the interior of X is connected. (If it makes any difference, suppose that the dimension n of X is 3.) I am interested in finding a minimal (or at least reasonably small) and preferably “nice” decomposition of X into … Read more

Are heat kernels on metric measure spaces continuous?

Let (M,d) be a separable, complete, compact metric space and μ a Radon measure with full support on it. Let E be a regular strongly local Dirichlet form on L2(M). There exists an associated self-adjoint non-negative operator, a strongly continuous semigroup and a heat kernel (or transition function) p(t,x,y). What I want to know is … Read more

Reach of manifold vs. CkC^k-manifold

The reach τM of a manifold M is the largest number such that any point at distance less than τM from M has a unique nearest point on M. This concept seems quite related to the local feature size introduced in computational geometry, but I haven’t seen the two compared. Two questions: Q1. Is the … Read more

Do homological holes with unit coefficients correspond to polyhedra?

(Originally posted at m.se without answers.) Let $T$ be a set of triangles in an abstract simplicial complex, with orientation of the triangles chosen such that $$\partial \left( \sum \limits_{t \in T} t \right) = 0$$ where the boundary operator is as usual, and we consider coefficients in $\mathbb{Z}$. Let $S$ be the support of … Read more