**1**

vote

**0**answers

18 views

### VC dimension of infinite cones

What is the VC dimension of infinite $d$-dimensional cones? ( single cones not double).
I would say $2d + 1$ or $O(d^2)$
Does anybody have any reference or ideas?

**-1**

votes

**0**answers

33 views

### conformal deformation with fixed boundaries

For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...

**0**

votes

**0**answers

20 views

### Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...

**4**

votes

**1**answer

140 views

### Ascertain properties of a new kind of rectilinear-convex set

PREABMLE TO MY QUESTION
I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many ...

**0**

votes

**0**answers

51 views

### Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that ...

**2**

votes

**0**answers

58 views

### Natural neighbor interpolation

Recently I am interested in Natural neighbor interpolation, that is :
Given a function $P(x)$ and some interpolation points $\{x_i,P(x_i)\}_{i=1}^N$, we have the interpolation function ...

**1**

vote

**0**answers

51 views

### Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb ...

**4**

votes

**0**answers

105 views

### Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...

**-2**

votes

**2**answers

74 views

### Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull) [closed]

Edit : Consider giving a reason for down vote.
In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of ...

**2**

votes

**0**answers

79 views

### Finding generators of symmetric cones

I have a bunch of vectors $\mathbf v_i$ in $\mathbb R^n$. I would like to consider the cone $C$ spanned by these vectors, together with all the other vectors that can be obtained by permuting the ...

**1**

vote

**0**answers

59 views

### Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...

**1**

vote

**0**answers

33 views

### Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = ...

**2**

votes

**1**answer

58 views

### Maximal opening angle of a polygon from a point [closed]

I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point.
One ...

**6**

votes

**1**answer

143 views

### Algorithm that generates a n-simplex that cover n-polytope?

Given an $n$-cube with unit volume, is there any algorithm that generates a $n$-simplex that covers the $n$-polytope?

**3**

votes

**1**answer

96 views

### 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?

**6**

votes

**2**answers

219 views

### Do computational geometers use Lagrange multipliers?

Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual ...

**0**

votes

**1**answer

62 views

### Detect perimetral edges of a polygon [closed]

I'm developing a building editor. Users can draw rooms by adding angles (vertices of the room) with a left click. Clicking on an existing angle closes the room and fills the floor by using the ...

**4**

votes

**1**answer

130 views

### Do random triangulation edge-flips maintain randomness?

Let $S$ be a fixed set of $n$ points in the plane in general position.
Let $T$ be a triangulation
of $S$, (somehow) selected
uniformly at random from all triangulations of $S$.
(There are an ...

**4**

votes

**1**answer

726 views

### Finding the vertices of a convex polyhedron from a set of planes

I'm new to computational geometry and advanced mathematics in general here so bear with me. I've spent a decent amount of time attempting to figure out this problem and I just can't find a solution.
...

**1**

vote

**0**answers

59 views

### Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy
$$
\left\|A-Id\right\|_{op}<r
$$
for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...

**4**

votes

**2**answers

323 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 ...

**5**

votes

**0**answers

134 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 ...

**4**

votes

**2**answers

71 views

### Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...

**5**

votes

**2**answers

578 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

**0**answers

221 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

**2**answers

292 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 ...

**7**

votes

**1**answer

179 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**4**

votes

**2**answers

127 views

### Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?

Is there an algorithm to construct a polyhedron containing all points in space for which there exists no ray to infinite not intersecting a given polyhedron?
In 2D, we could consider polygons. For ...

**4**

votes

**1**answer

258 views

### Finding the “top” or “bottom” vertex of a simplex

A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v_1 \ge p_1$, ... , $v_n \ge p_n$). Similarly, $v$ is a "bottom" ...

**5**

votes

**3**answers

2k views

### Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space?
I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a ...

**3**

votes

**1**answer

237 views

### Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...

**2**

votes

**2**answers

1k views

### Trilateration problem

When trying to develop an algorithm for a program, I got with the following problem:
Determine the approximate location of $O$, if you can take finite samples $P_n$ from known locations and always ...

**1**

vote

**1**answer

172 views

### Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier

Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...

**9**

votes

**3**answers

748 views

### Is every graph an edge-crossing graph?

Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph ...

**11**

votes

**0**answers

155 views

### Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...

**2**

votes

**0**answers

93 views

### The intersection of two $l_1$ balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap ...

**6**

votes

**1**answer

136 views

### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...

**2**

votes

**0**answers

113 views

### Intersecting balls with convex regions and a bisector thereof

This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...

**1**

vote

**0**answers

46 views

### Concise disambiguation of Voronoi boundaries

Say that $x_1,\dots,x_n$ are points in the plane, with a Voronoi diagram $V_1,\dots,V_n$. The Voronoi diagram is typically defined by $$V_i = \{x:\|x-x_i\|\leq \|x-x_j\|~\forall j\}~.$$ Is there any ...

**2**

votes

**0**answers

96 views

### Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)

Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of ...

**2**

votes

**3**answers

882 views

### Is there a simple criterion to determine if two parallelograms intersect?

Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.

**0**

votes

**1**answer

100 views

### Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)

I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...

**24**

votes

**4**answers

4k views

### Computational software in Algebraic Topology?

I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example:
Create a simplicial complex/set and ask questions about its ...

**0**

votes

**2**answers

317 views

### algebraic topology and 3d/4d printing [closed]

I googled for papers on applying algebraic topology to 3d/4d printing. It just seems to me that there has to be a connection. Any help, kind audience?
edit: 4d printing means 1-parameter families of ...

**3**

votes

**2**answers

112 views

### Average vertex degree in finite Delaunay triangulations in high dimensions

In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6.
Does anyone know of a similar result in higher dimension? I ...

**1**

vote

**0**answers

36 views

### Determining feasibility of specific intersection structures between closed paths in the plane

I have two arrays, each containing a different ordering of the same set of integers. Each integer is a label for a point in which two closed paths intersect in the plane. The two arrays are ...

**2**

votes

**1**answer

99 views

### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...

**2**

votes

**1**answer

60 views

### What is Known about Preprocessing for Stabbing Queries?

In a concrete setting, I have the following problem:
given a fixed set of simple objects (e.g. disks or, convex polygons with few vertices), I need to quickly report the objects that are hit (i.e. ...

**2**

votes

**1**answer

504 views

### Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...

**14**

votes

**1**answer

338 views

### Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...