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4 votes
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Length and curvature for closed curves in negatively curved spaces

The Reshetnyak majorization theorem (see 9.56) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be majorized by a convex plane figure $F$; that is, there is short (= ...
0 votes

Packing twelve spherical caps to maximize tangencies

As an extension of the problem (which turns out to have some physical significance), we may consider what would happen if we were to allow the twelve vectors to "land" within the unit ball ...
1 vote

Cone unfolding of space curves

Pardon me for this bit of self-promotion, especially because this is only tangential to the OP's concerns. But cone unrolling and Anton Petrunin's mention of Alexandrov's developments, in conjunction ...
1 vote

Cone unfolding of space curves

Liberman used cylinder unfolding to study geodesics on convex surfaces. [Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.] Right now standard ...
1 vote

4D Duoprisms based on nonconvex polygons

The great duoantiprism can be constructed using a compound of two pentagonal-pentagrammic duoprisms by inserting additional edges, or by alternating a decagonal-decagrammic duoprism.
0 votes

Which pyramids fill space?

About the tetrahedra problem, I have found this reference/proof that they cannot be used to fill 3D space (since the proven upper bound is smaller than $100 \cdot (1-2.7 \cdot 10^{-25})\%$: https://en....
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0 votes
Accepted

Finitely generated groups with Hölder-exotic space of ends?

The question is solved positively in the paper The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups by Matthieu Dussaule, Wenyuan Yang, which appeared on arXiv on October ...
2 votes
Accepted

Intersection of conical neighbourhoods on a polyhedral space

You say "The same is true for a tubular neighborhood of the edge". This is not correct, but it is true if you stay away from the endpoints. So $U$ should be defined as a tubular neighborhood ...
0 votes

If M times circle admits a locally CAT(0)-metric, then M also carries a locally CAT(0)-metric?

Here is a partial answer, I will prove the following: If $M\times \mathbb{S}^1$ equipped with locally $\mathrm{CAT}(0)$ length metric, then there is isometric $\mathbb{S}^1$-action on $M\times \...
2 votes

Tangent cone of a proper CAT(0) is a proper CAT(0) space

The answer is "yes" for geodesically complete CAT(к) spaces. It follows directly from the comparison.
3 votes

m-point-homogeneous, but not (m+1)-point-homogeneous

The Schläfli graph, viewed as a metric space, is 4-homogeneous but not 5-homogeneous.
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2 votes
Accepted

Spaces satisfying a strong Cartan-Hadamard theorem

Note that Hilbert spaces (of all dimensions finite or infinite) are the only geodesic spaces with extendable geodesics which are flat in the sense of Alexandrov. Therefore $X$ has to have extendable ...
1 vote

Distance of average of points to center of minimum enclosing ball

Assume $d=2\cdot k$. If $d$ is large, then the vertices of the cube such that $\|v\|=k$ lie very densely in $(d-1)$-dimensional sphere. If $n$ is large, but not that large, then you may choose $n-1$ ...
2 votes
Accepted

Expected doubling constant of a random Erdős–Rényi graph

Let me assume $p > (1 + \varepsilon)(2 \ln n/ n)^{1/2}$, this in particular includes the case of constant $p \in (0,1)$. If $p > (1 + \varepsilon)(2 \ln n/ n)^{1/2}$ then w.h.p. the binomial ...
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1 vote

Are there infinitely many "generalized triangle vertices"?

I found this property of X(1138) no later than 2015. I called them X(4),X(74) and X(1138) system centers and tried to find another one but failed. I also found they all lie on the Neuberg's cubic ...
0 votes

For which metric spaces is Gromov-Hausdorff distance actually achieved?

In the embedding problem we care only about the distances between the images of $x\in X$ and $y \in Y$ denoted by $\rho(x,y)$, the rest of the container space being superfluous. Therefore, we consider ...
3 votes
Accepted

Extending a partially defined metric on a metrizable space

Here is a counterexample to Q2, with your stated extra condition. Let $X$ consist of the half-open unit interval $(0,1]$ on the $x$-axis in the plane, together with the full unit interval $[0,1]$ at ...
2 votes

When is the angular metric on the space of directions intrinsic?

One may ask if locally compact Alexandrov spaces have intrinsic spaces of directions --- I do not know the answer. See 13.40 in our book.
1 vote

Is the affine geometry a geometry of proportions?

The midpoint of $a$ and $b$ can be defined from equiproportion as the unique $m$ for which $(a,m,b)$ and $(b,m,a)$ are equiproportional. Similarly, the collinearity of $a,b,c$ can be defined as one of ...
  • 18k
4 votes

Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

It is, indeed, true that if we have a polygon that can be partitioned into $120^\circ$-obtuse triangles, then at least one angle of the original polygon exceeds $120^\circ$. However we may need extra ...
  • 54.3k
7 votes
Accepted

Is there a way to quantify the chirality of a 3d shape?

This is a topic of some research, summarized in On quantifying chirality — Obstacles and problems towards unification. One metric is the Hausdorff chirality, which quantifies the chirality of a ...
2 votes

Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles

It is not the case that all strongly obtuse subdivisions with a threshold over $120^\circ$ are possible through vertex-to-vertex connections. Consider the following dissection of a pentagon into ...

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