Questions tagged [metric-geometry]

Question 1

Let $f:M\to\mathbb{R}^k$ be smooth, where $M$ is a closed manifold of dimension $n>k$. I am thinking about the continuity properties of the map $c\mapsto\mathcal{H}^{n-k}(f^{-1}(c))$. Clearly this ...

Question 2

Let $X$ be a 2-dim CBB($k$) space, and $AO,BO,CO$ are all geodesics, do WE have $$\angle AOC=\angle AOB+\angle BOC\,?$$ I know that if $AC$ is geodesic, then $\angle AOB+\angle BOC=\pi$, but in ...

Question 3

Some context and motivation for this question: A friend posed to me the following puzzle. Given eight points in the unit disc, why must there exist a pair of points with distance less than $ 1 $? The ...

Question 4

I'm studying Roman Vershynin's High-Dimensional Probability and am working through Corollary 0.0.4, where he constructs an epsilon-net for a polytope $P$ via averages of vertices. Here's what confused ...

Question 5

A set $E \subset \mathbb{R}^d$ is called $1$-rectifiable if there exists a (countable) family of Lipschitz mappings $f_i : \mathbb{R} \rightarrow \mathbb{R}^d$ such that $$ H^1 \bigg(E \setminus \...

Question 6

Let $M$ be a 3-dimensional Cartan-Hadamard manifold (complete, simply connected, with nonpositive sectional curvature). Let $S \subset M$ be a $C^{1,1}$ surface enclosing a domain $E$, and let $D_0$ ...

Question 7

Let $M$ be a 3-dimensional Cartan–Hadamard manifold (complete, simply connected, nonpositive sectional curvature), let $S \subset M$ be a $C^{1,1}$ surface enclosing a domain $E$, and let $D_0$ be the ...

Question 8

Let $(M,g)$ be a Cartan-Hadamard manifold (simply connected, complete and with sectional curvature K≤0 ) and let $K \subset M$ be a closed convex body with $C^{1,1}$ boundary. Fix an interior point $p ...

Question 9

While studying the notion of asymptotic dimension, I got interested in the examples of classes of metric spaces with a computable asymptotic dimension. The definition of asymptotic dimension I'm used ...

Question 10

I have a convex curve $\gamma$ on a two-dimensional Riemannian manifold with positive Gaussian curvature, or a Alexandrov space with curvature $\geq 0$, which is a simple closed curve, and the region ...

Question 11

I was studying the following discussion on Mathoverflow: Continuous automorphism groups of normed vector spaces, where it is mentioned that "The isometry group of any (real) finite-dimensional ...

Question 12

For a closed convex curve $\gamma: I \to \mathbb{R}^2$, where the region enclosed by the curve is convex, the Menger curvature is defined as follows: for three distinct points $m, p, q$ on the curve, ...

Question 13

Theorem in question: I understood the first part, I don't understand the converse. Why can we assume that for all $v \in V_{ij}$ point $w_i$ is the nearest neighbor? Can't we say that $w_i$ or $w_j$ ...

Question 14

Im struggeling to do a proof of Gromov-Bishop volume comparison theorem restricted to the 2 dimensional case. I have very little experience in working with Riemannian geometry, therefor the need to ...

Question 15

I have recently come across the following, very intuitive, statement within the book "A course in Metric Geometry" by Burago, Burago, Ivanov. It asserts that we can calculate the length of ...

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Questions tagged [metric-geometry]

Smooth functions with arbitrarily large regular level sets

Angle problems in two-dimensional CBB

Can the unit disc be covered by seven squares, each of area one-half?

Understanding the combinatorial bound choice in a covering argument (Vershynin's HDP)

Rectifiable Subsets of Sets with Large Hausdorff Dimension

Classical comparison theorems for geodesics starting from a submanifold whose sectional curvature is bounded from below

Nonpositive sectional curvature implies nonnegative mean curvature on convex boundary?

Do geodesics from an interior point intersect the boundary of a convex body only once?

Examples of metric spaces with computable asymptotic dimension

Does the Lipschitz mapping preserve angles?

Isometry Group of a Finite-dimensional Normed Linear Space

The existence of the Menger curvature of a closed convex curve.

How does the existence of a second order Voronoi polyhedron imply corresponding first order Voronoi polyhedra to be adjacent?

Proof of Gromov-Bishop theorem in a 2D setting

Calculating the Length of a (non-simple) Curve via the Hausdorff Measure of Cardinalities of Preimages

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