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

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2
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0answers
37 views

The most number of points that realize only $k$ distinct distances

For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$ in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances, $||p_i-p_j||$. Example. For points in the plane ...
15
votes
2answers
439 views

Is every closed curve in 3D a geodesic on a genus-0 surface?

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$. Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a ...
5
votes
0answers
98 views

Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell

Given an $n \times n$ grid with unit grid cells, and one point from the interior of each cell, what is are best possible lower and upper bounds for lengths of minimum spanning trees? The lower bound ...
8
votes
11answers
3k views

Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry ...
3
votes
0answers
527 views

What is known about the area of the symmetric Pythagorean tree?

What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to ...
2
votes
0answers
143 views

Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...
2
votes
0answers
45 views

Euclidean minimum spanning trees intersecting each unit square

The recent question "Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell" can be restated in terms of "minimum spanning trees intersecting each (closed) lattice square of an ...
23
votes
5answers
2k views

What is a good method to find random points on the n-sphere when n is large?

As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at ...
9
votes
1answer
461 views

What is the shape of the $n$-gon which gives the maximum of a function?

What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by $$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i ...
-3
votes
0answers
12 views

Descrition of clipping algorithm in Murta's gpc [on hold]

I have searched but failed to get a algorithmic description of the algorithm used by Alan murta in general polygon clipper.It is NOT vatti,for sure. Unfortunately old versions of his code are also ...
2
votes
1answer
22 views

Algorithm to find the vertices of the equidistant lines between N closed polygonal lines

I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by ...
2
votes
0answers
95 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$. Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
18
votes
5answers
865 views

Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the question, "Can the unsolvability of quintics be seen in the geometry of the icosahedron?", that There is not a single ...
33
votes
6answers
3k views
8
votes
5answers
725 views

covering by spherical caps

Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...
4
votes
1answer
108 views

Another 2D theorem which generalizes to higher dimension

I have found a new property of the Tetrahedron. In fact, this is a 3D generalization of theorem 1 in my paper "A Note on Reflection", published by Forum Geometricorum. Consider any Tetrahedron ABCD. ...
5
votes
1answer
98 views

Pigeonholing Polygons: Can two rigid regions fit in twice the space needed?

This is a tweak of Henry Segerman's question Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1? , but restricted to the point of possibly having a proof in the ...
0
votes
1answer
80 views

Extreme points of convex hull of Minkowski sum [closed]

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_n\}$. Let $P = \operatorname{conv}(a_1,\ldots,a_p)$ and $Q = \operatorname{conv}(b_1,\ldots,b_q)$ be two convex sets ...
4
votes
3answers
952 views

Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
13
votes
2answers
745 views

Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
8
votes
2answers
465 views

When does every point in a polytope lie along a chord between its edges?

Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...
4
votes
1answer
269 views

Generalizing the circle packing theorem to 3-dimensions

The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, ...
15
votes
1answer
2k views

A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...
5
votes
2answers
1k views

What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
6
votes
1answer
431 views

Using mirrors to make a non-convex polygon visible from a fixed interior point

Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
3
votes
1answer
229 views

Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
12
votes
2answers
448 views

A problem on convex geometry

Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are ...
2
votes
1answer
94 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
1
vote
3answers
158 views

Isometric imbedding of finite metric space into standards spaces [duplicate]

Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space? (For $n=3$ this is true.) If not, what are ...
2
votes
1answer
181 views

Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic: The ...
22
votes
1answer
412 views

Expected number of vertices of a hypercube slice — is this new/interesting?

I am a (mostly) amateur mathematician, but my education and work have featured a lot of mathematics, and recently I bumped into a mathematical problem for which I can find no references, and I am ...
16
votes
1answer
348 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
19
votes
2answers
910 views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
5
votes
2answers
494 views

Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...
2
votes
0answers
196 views

Find minimum-area ellipse enclosing a set of ellipses, all centered at the origin

Given a set of N > 2 (two-dimensional and coplanar) ellipses, all centered at the origin, how do I find the ellipse with the minimum area which encloses all of them? Background: Thanks to Will Jagy ...
4
votes
2answers
207 views

Breaking a rectangle into smaller rectangles with small diagonals

Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
1
vote
0answers
36 views

Projection of a ray onto a random polytope

Suppose $P$ is a polytope formed by $p$ (general) random planes in $\mathbb{R}^n$. We assume $p \asymp n$ and $P$ has a diameter $O(\sqrt{n})$. For any $x \in \mathbb{R}^n$, denote by ...
3
votes
1answer
307 views

On distances between points on the plane

Take a set of $2n$ points in the plane and assume that no open set of diameter $1$ contains more than $n$ of these points. Question: can we pair up the points so that the distance between the points ...
4
votes
2answers
211 views

A (possibly boring) Voronoi Game

The board for this game is a compact convex region $\cal C$ of $\mathbb{R}^2$. Below I illustrate with $\cal C$ an equilateral triangle. Two players, $A$ and $B$, alternate turns. At each turn they ...
4
votes
2answers
236 views

What are the applications of Voronoi diagrams in pure mathematics? [closed]

Voronoi diagrams have interesting mathematical properties and applications in algorithms and modeling. But what are its applications in pure mathematics? For example, what theorems can be proved using ...
19
votes
1answer
437 views

Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems on convex polyhedra. Progress has been made on several of his problems, and perhaps some have been completely ...
0
votes
1answer
189 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
2
votes
2answers
168 views

4-polytope with vertices at the binary octahedral group

Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identitfying $H$ with $R^4$). The binary tetrahedral group lies at the vertices of the ...
6
votes
1answer
173 views

Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) , I started wondering if the center of gravity is always contained in the closed convex hull. More precisely, given ...
10
votes
2answers
246 views

Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
3
votes
1answer
150 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
3
votes
0answers
28 views

Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
7
votes
0answers
78 views

What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a product of trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ ...
5
votes
1answer
229 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
8
votes
1answer
307 views

Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...