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

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2
votes
1answer
64 views

Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it. Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...
10
votes
3answers
323 views

Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$) of radius $r < 1$ avoid all rational points of height $\le h$? A rational point is a point all of whose coordinates ...
5
votes
2answers
208 views

Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$ (although my question may as well be asked for the surface of a polyhedron). Say that $\gamma$ is a shortest halving curve if (a) it partitions the ...
15
votes
0answers
427 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
2
votes
0answers
87 views

Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
3
votes
2answers
210 views

Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
-3
votes
2answers
96 views

Hexagon Formed by connecting Trisections of triangle sides [closed]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
10
votes
2answers
488 views

Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$, there exists some graph $G$ whose every embedding into $\mathbb{R}^3$ (or into $\mathbb{S}^3$) contains a cycle that realizes $K$? I know the ...
27
votes
2answers
754 views

Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
3
votes
1answer
119 views

Are there isomeasure simplices?

Say that two polyhedra in $\mathbb{R}^3$ have isomeasures (my terminology) if they have: the same volume, the same surface area, the same sum of all edge lengths, and the same number of vertices. The ...
23
votes
2answers
624 views

Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...
3
votes
0answers
77 views

maximum sum of angles between $n$ lines

Take $n$ lines in $\mathbb{R}^d$ (not necessary different, and all passing through the origin, though this is not important). What is maximal possible sum of angles between them for given $n$ and $d$? ...
3
votes
0answers
42 views

Algorithm to construct metric space endomorphism with controlled square

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be ...
13
votes
0answers
240 views

Can R^3 be expressed as a disjoint union of pairwise linked circles?

We can express $\mathbb{R}^3$ as a disjoint union of circles. There are some constructive ways of doing this, although it's easier to construct them sequentially by transfinite induction, applying the ...
2
votes
1answer
139 views

Distance matrices

We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x_1,\cdots,x_n \in X$ such that $M=[d(x_i,x_j)]_{n\times n}$. Question. For which ...
4
votes
2answers
123 views

Connectivity of points sampled in a grid

Suppose that I partition an $n\times n$ square into $n^2$ squares $S_1 ,\dots, S_{n^2}$ each of area $1$, and then I sample a point $X_i$ uniformly at random in each $S_{i}$. Now fix a radius $r$ and ...
3
votes
1answer
243 views

When does the intersection of cylinders produce a ball?

Suppose one intersects unit-radius solid cylinders in $\mathbb{R}^3$, with each cylinder axis passing through the origin. For example, two such cylinders produce the Steinmetz solid. But if we ...
2
votes
0answers
113 views

On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...
4
votes
0answers
84 views

Centralizers and intersections in the Gromov-boundary of the mapping class group

The mapping class group of a punctured surface $\Sigma$ is weakly relatively hyperbolic (see below), hence it is well defined the Gromov-boundary with respect to the relative metric. First question: ...
1
vote
0answers
95 views

A conjecture about cross sections of a pyramid [closed]

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a conjecture I came across while trying to solve this problem. I was ...
3
votes
1answer
65 views

Finding a minimum covering of a polygon with interesting shapes

After reading many papers about problems of minimum polygon covering, I found out that there are four different types of units that are considered for covering polygons, in increasing order of ...
8
votes
1answer
403 views

Polyhedron not circumscribed about a sphere

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere. My teacher ...
2
votes
0answers
57 views

isoperimetric problems on Alexandrov spaces

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by $$ ...
4
votes
0answers
157 views

Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk and a point directly above the disk center at height $h$.                 Is the ...
6
votes
1answer
134 views

Hiding $k$ disks inside a larger disk

Suppose one has $k$ unit-radius disks, and the goal is to hide them inside a disk of radius $R \gg k$. The detection probes are rays along a line. (Think of the disks as tumor cells, and the rays as ...
1
vote
0answers
130 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
4
votes
1answer
177 views

Light inside a polyhedron

I have two questions the same as Mostafa's Question: Visibility of vertices in polyhedra Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ ...
19
votes
3answers
771 views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
1
vote
0answers
56 views

Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface. (You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
8
votes
1answer
195 views

Billiard dynamics with angle of reflection a fraction of angle of incidence

Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting in a mirrored square) has the property that the angle of reflection is a fraction of the angle of incidence, rather ...
2
votes
1answer
75 views

Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
0
votes
0answers
89 views

Geometric interpretation of table with permutations and inversions

Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg. $n=1,...,6$, ...
3
votes
1answer
243 views

Taylor expansion of the determinant of a Riemannian metric

Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...
2
votes
0answers
165 views

Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...
3
votes
1answer
220 views

Characterization of $l_p$ up to a linear isometry

There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach ...
14
votes
3answers
644 views

“Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ ...
2
votes
1answer
93 views

Bound on maximum distance between points on a unit N-Sphere

I want to select M points on the N-sphere such that $min_{i\neq j,i,j\in \{1..M\}} ||x_i - x_j||$ is maximized. Are there good upper bounds for this max-min distance?
3
votes
0answers
168 views

Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I ...
3
votes
0answers
68 views

Covering fat objects with fat objects

The family of rectangles has the cover property, i.e.: For every $R\geq 1$, $k\geq 1$: every rectangle with aspect ratio $kR$ can be exactly covered by $\lceil k\rceil$ (possibly overlapping) ...
0
votes
1answer
107 views

both convex and superharmonic function on manifold concave?

M is a non-compact Rimannian manifold without boundary. $f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e. $$ -\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M ...
7
votes
2answers
338 views

Riemannian distance functions on the real line

A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties: $d$ is a length metric ...
13
votes
0answers
345 views

Surface area of convex hull [duplicate]

Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
3
votes
1answer
143 views

Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?

Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$. Assume that $S$ satisfies ...
2
votes
1answer
112 views

The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to ...
2
votes
2answers
114 views

Volume of specials sets on sphere $S^N$

Suppose I'm given $m$ points $\{q_i\}$ on the sphere $S^N$. I want to get a lower/upper bound for the volume of the following sets with respect to uniform probability measure $\mathbb{P}$ on the ...
7
votes
0answers
229 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
22
votes
0answers
376 views

Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
1
vote
1answer
34 views

Maximum crossings of curvature-constrained curve

Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$. If $C$ has length $L$, what is the largest number of proper self-crossings of $C$ as a function of $L$? For example, the curve ...
10
votes
1answer
551 views

On bi-invariant metrics on groups

I'm looking for a quick, snap-your-fingers proof of the following result: A continuous length metric on $\mathbb{R}^n$ that is invariant under translations comes from a norm. To be clear about ...
16
votes
0answers
228 views

“Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations?

A beautiful theorem known as the Japanese Theorem (Wikipedia, MathWorld) says that, no matter how one triangulates a cyclic (inscribed in a circle) polygon, the sum of the radii of the incircles is ...