# Questions tagged [mg.metric-geometry]

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

2,339 questions
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### Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
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### Sphere packing in a sphere

Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number ...
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### How to compute the parameters of circumscribed hypershpere? [on hold]

Assume I have an $n$-dimensional simplex on the points $x_0, ..., x_n$ where each $x_i \in \mathbb{R}^n$. I would like to obtain the parameters (center and radius) of it's circumscribed n-dimensional ...
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### Metrically Ramsey ultrafilters

On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
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### Sums of squared distances between points on an $n$-sphere

I have discovered the following results about the sums of squared distances between points on an $n$-sphere (and proved them). To the best of my knowledge (and my advisor's knowledge), these results ...
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### Gluing hexagons to get a locally CAT(0) space

I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space: The first two give the torus and the Klein bottle, respectively. What are the ...
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### Bishop-Gromov inequality strengthened for anisotropic metrics?

The Bishop-Gromov inequality provides an upper-bound on the rate of growth of volume of a ball of radius $r$ in spaces that have a lower-bound on the Ricci curvature, $Ric \geq (n-1)K$. (I am ...
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### What is the meaning of Conjugate radius and Injectivity radius?

I review the text book of differential geometry and I find that the conjugate radius and injectivity radius are still enigmatic for me. Here is a quetion which I confuse it. I don't think it is true, ...
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### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from ), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...
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### Convex sets in Alexandrov spaces

Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary? Definition of a convex subset: $A\subseteq X$ ...
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### Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...
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### Term for a metric space for which the triangle inequality is strict?

Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity. For instance, a subset of ...
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### Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
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Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by $$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |... 0answers 80 views ### Estimation of the Gromov–Wasserstein distance of spheres Let (X,d_X,\mu_X) and (Y,d_Y,\mu_Y) be two metric measure spaces. A probability measure \mu over X\times Y is called a coupling if (\pi_1)_\sharp \mu=\mu_X and (\pi_2)_\sharp \mu=\mu_Y. We ... 0answers 447 views ### Disc bounded by a plane curve Let \Sigma be a sphere topologically embedded into \mathbb{R}^3. Is it always possible to find a disc \Delta\subset\Sigma which is bounded by a plane curve? It is easy to find an open disc ... 0answers 965 views ### Sofa in a snaky 3D corridor What is the largest volume object that can pass though a 1 \times 1 \times L "snaky" corridor, where L is large enough to be irrelvant, say L > 6. ... 0answers 66 views ### Equal volume and projections Given three unit vectors u_1,u_2,u_3 in \mathbb{R}^3, can we find some body K \subset \mathbb{R}^3 (probably convex) such that the following three things hold (1) |P_{u_1^\perp}K|=|P_{u_2^\... 1answer 399 views ### Axiom of choice and a set in the plane that intersects every line in two points In this question Subset of the plane that intersects every line exactly twice someone ask for a reference of a paper where they proof the result : ''There exist a subset of the plane that intersects ... 1answer 61 views ### Probability of two Points being divided by an high-Dimensional Hyperplane I have two points x_1,x_2 \in \mathbb S^n  which are distant d from each other, where d<<1. I also have a vector v sampled uniformly at random from \mathbb S^n. What is the ... 0answers 139 views ### A special connected subset of the Cantor fan Is there a dense connected subset X of the Cantor fan$$(C\times [0,1])/(C\times \{1\})$$such that for every two connected subsets X_1,X_2\subseteq X, the intersection X_1\cap X_2 is connected? ... 0answers 86 views ### Locally compact metric spaces whose group of isometries generated by isometries of small displacement is transitive enough I wasn't sure how to make the title any more precise than that. Let (X,d) be a locally compact metric space. For any \varepsilon>0 let \mathrm{Aut}_\varepsilon (X) be the subgroup of the ... 0answers 38 views ### Modulus of image of a curve family in a rectangle I don't expect to get a positive answer to this question but I may as well try. Let R be the rectangle in \mathbb{C} given by \{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\} for some l,h>0. ... 11answers 3k views ### Creating high quality figures of surfaces I am not sure if this question is suitable for mo, it is more about visualization than math. Anyway, here it is: What is the best way to visualize a 2-surface in Euclidean space with high quality? ... 2answers 111 views ### Reference request: \alpha-Hölder spaces as double duals If (X,d) is a complete metric space, we define the \alpha-Hölder class \Lambda_\alpha(X) as the subset of C_b(X) satisfying that$$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$... 1answer 47 views ### parametrize triangles meeting certain conditions Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable. 0answers 53 views ### Homeomorphism type of the horofunction boundary for nilpotent Lie groups Consider a metric space (X,d) and fix a base point w. A horofunction is a function of the form$$\beta_y(x)=d(x,y)-d(w,y). The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...
It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ...