Tagged Questions

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

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Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
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Pairs of rays in euclidean buildings

In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...
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$|\exp_p(x)\exp_q(T(x))|$ controlled by $|pq|?$ $T$ is parallel transportation in Alexandrov space

Let $M$ be an Alexandrov space with $sec \geqslant 0$. Let $p$ and $q$ be points of shortest path $\gamma$ in $M$, that are not end point. Then the tangent cone can split as $C_p=L_p \times \mathbb{R}$...
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How does the singular surfaces obtained when the border of a Euclidean set becomes a point look like?

I'm curious to understand in several manner, what is the metric geometry of the metric space homeomorphic to a sphere, obtained from a compact convex set $K\subset R^2$ with the Euclidean distance, ...
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The circle with minimal radius covering known finite set of points on a plane

Given some points on a plane, how to determine the circle with minimal radius covering all these points?
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Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...
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Is there a subset of the plane that meets every line in two open intervals?

Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
This suggests a variant on Polya's orchard problem. That problem asks1 for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...
$C^1$ regularity of harmonic functions on Riemannian manifolds
Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$. I'm interested in knowing whether there ...