All Questions
1,544 questions
1
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28
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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
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0
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65
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To partition a triangle into $n$ convex pieces with sum of number of sides over all pieces maximized
This post is a variant on To cut a triangle into $n$ $p$-sided polygonal regions.
Question: Given a positive integer $n$, a triangular region is to be cut into $n$ convex pieces so that the sum over ...
- 5,979
-1
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0
answers
41
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Is it possible to backtrack an optimization solver? [closed]
I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
2
votes
1
answer
128
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Reference request for elementary convex geometry property
I need to use the following lemma for a proof. It is an elementary result, which I am sure is well known, just I am not familiar enough with the relative literature to find a direct reference. Is ...
- 345
3
votes
0
answers
31
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Disjoint touching bodies of constant width
Let $F$ and $F_1,\ldots,F_n$ be bodies of constant width 1 in $\mathbb{R}^d$ such that $F_1,\ldots,F_n$ are pairwise disjoint and all intersect non-trivially (i.e. in at least one point) with $F$. ...
- 189
2
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0
answers
85
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On the trajectory followed by a point P on a planar convex region C when P is mapped repeatedly to the farthest point to it on C
Consider a planar convex region $C$. Let us define a mapping of a point $P$ on $C$ to that point on C that is farthest from $P$. Obviously, if from an initial position of $P$, we do this mapping ...
- 5,979
2
votes
1
answer
91
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Does approximately null gradient imply approximately global minimum for convex functions?
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
- 225
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0
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35
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Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?
Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
- 1,465
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0
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21
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Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
3
votes
0
answers
122
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Helly number of closed unit balls in R^d
Does anyone know the Helly number of closed unit balls in $\mathbf{R}^d$ and a reference? This number is the smallest number $k(d)$ so that for any collection of of closed unit balls in $\mathbf{R}^d$,...
- 189
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26
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How to calculate the vertices of a convex polytope (k-DOP)
I am currently reading Christer Ericson's Real-Time Collision Detection Book. The topic I'm particularly interested in, is the chapter about Discrete-orientation Polytopes (k-DOPs). In his words "...
- 1
2
votes
1
answer
168
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Ratio of inscribed/circumscribed ellipsoids: geometrical proof?
Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
- 52.3k
1
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37
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Metric entropy of an ellipsoid
Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map.
Consider the function
$$
H(T) := \log M(TB_2^d, B_2^d),
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$....
- 460
3
votes
1
answer
136
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For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
- 1,955
7
votes
2
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242
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Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
-1
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0
answers
64
views
Axes of symmetry and symmetry group of the tangent cone to an open, connected, convex subset of the Euclidean space
Given a closed convex set $K\subset \mathbb{R}^d$ and a point $x\in K$ the tangent cone to $K$ at $x$ is defined by
\begin{equation}
T_xK:=\overline{\{v\in \mathbb{R}^d: \exists \lambda \geq 0 \text{ ...
- 1,465
1
vote
0
answers
118
views
'Uniformity' of surfaces of 3D convex solids
We try to go a little further from Which convex solids have geodesics on the surface that lie entirely in a plane?
Definitions: The surface of a finite 3D convex body may be called a convex surface. ...
- 5,979
0
votes
0
answers
36
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
- 101
6
votes
1
answer
346
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Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
- 93
1
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0
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72
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Existence of a sequence of $-1/1$-polytopes with certain geometric properties
Let $P_n \subset \mathbb{R}^n$ be a sequence of polytopes (A polytope is the convex hull of finitely many points). Let $B_n \subset \mathbb{R}^{n}$ denote the Euclidean unit ball. I am interested in ...
- 93
0
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0
answers
27
views
Projection onto polytopes as tropical polynomial
Let $C$ be a convex polytope in $\mathbb{R}^n$ with $m$ extremal points. Let $p\in \{1,2\}$.
Can the $\ell^p$-projection $\Pi_C:\mathbb{R}^n\to C$
$$
\Pi_C(x) \in \operatorname{argmin}_{z\in C}\, \|x-...
- 364
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157
1
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0
answers
99
views
Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,149
1
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0
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109
views
Which convex solids have geodesics on the surface that lie entirely in a plane?
We add a bit to On partitioning the surface of a convex solid into geodesically convex equal area regions
Consider a convex 3D solid body C - not necessarily a polyhedral body. What could be said ...
- 5,979
1
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1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
7
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1
answer
290
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Is it necessarily true that the maximal section of a centrally symmetric convex body is always bigger than its minimal projection?
I hope everyone is doing well.
Let $K \subset \mathbb{R}^n$ be a centrally symmetric convex body $(K = -K)$. Denote by $K \mid H$ the orthogonal projection of $K$ onto $H$, where $H$ is an $n - 1$ ...
- 83
0
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0
answers
69
views
Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces
Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
- 1,465
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0
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67
views
Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 460
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0
answers
21
views
Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
7
votes
0
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315
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
1
vote
0
answers
71
views
Is the circumcenter Lipschitz on large convex sets in hyperbolic space?
Given a uniquely geodesic metric space $X$, let $\mathcal K(X)$ denote the metric space of compact, convex subsets of $X$ equipped with the Hausdorff distance. Given $K \in \mathcal K(X)$, let $c(K)$ ...
- 498
2
votes
0
answers
70
views
Lipschitz continuity of orthogonal projection with respect to the Hausdorff distance
Let $x_0 \in \mathbb R^n$, and let $\mathcal K$ denote the set of compact convex subsets of $\mathbb R^n$ equipped with the Hausdorff metric. Consider the map $f: K \mapsto \Pi_K x_0$, where $\Pi_K$ ...
- 498
1
vote
0
answers
143
views
Integer points inside the high-dimensional ball (asymptotics)
Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:
$$...
- 435
5
votes
2
answers
112
views
Is there any equivalence between standard d dimensional Gaussian surface measure and d dimensional Hausdorff measure on boundary of convex sets?
I am currently going through the papers of Nazarov (2003): "On the maximal perimeter of a convex set in $\Bbb R^n$ with respect to a Gaussian measure" (MR2083397, Zbl 1036.52014) and Ball (...
3
votes
1
answer
237
views
Find the number of triangles in plane
Let $S$ be a set of $n$ points in the plane in general position. Each 3 points of S span a triangle. Total number of triangles spanned by S:
$$\binom{n}{3}=\frac{n(n-1)(n-2)}{6}=\frac{1}{6} n^3-O(n^2 )...
- 214
1
vote
1
answer
132
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Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
1
vote
0
answers
45
views
Inequality Involving Concave Monotonic Function
Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
0
votes
0
answers
45
views
Support function of the intersection of two $\ell_p$ balls
Denote $\|\cdot \|_p$ for the norm in $\ell_p^n$, where $1 \leq p \leq \infty$, and $n \geq 1$.
Let $(x^\star_i)$ denote a nonincreasing arrangement of the sequence $(|x_i|) \in \mathbb{R}^n$.
We ...
- 460
4
votes
1
answer
356
views
Left and right halves of convex curve
Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even.
A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points ...
- 214
3
votes
0
answers
110
views
How many Tverberg partition are in cloud of points? [closed]
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
- 214
0
votes
0
answers
48
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
6
votes
1
answer
366
views
An arrangement of hyperplanes [closed]
An arrangement of hyperplanes in $\mathbb{R}^d$ is simple if the hyperplanes are in general position (for every $1\leq k\leq d+1$, the intersection of $k$ hyperplanes is $(d-k)$-dimensional).
My ...
- 214
2
votes
0
answers
104
views
Reference request: books on convex analysis / geometry
I am interested in convex geometry and analysis, especially in its connections with high dimensional probability theory.
I was reading the book by Pisier, The volume of convex bodies and Banach space ...
- 460
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 988
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 364
2
votes
0
answers
63
views
Convex planar regions such that every boundary point has a 'fair bisector' passing thru it
We add a little to On 'fair bisectors' of planar convex regions and A claim on the concurrency of area bisectors of planar convex regions .
A fair bisector of a planar convex region is a line ...
- 5,979
0
votes
0
answers
82
views
On 'Bisecting sections' of 3D convex bodies
Following shadows and planar sections, we ask about bisecting sections. This post also continues Convex planar regions with all area bisectors having equal length and A claim on the concurrency of ...
- 5,979
6
votes
1
answer
127
views
Convex planar regions with all area bisectors having equal length
Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries.
An area bisector (perimeter bisector) of a planar convex region is a chord ...
- 5,979
1
vote
1
answer
103
views
Algorithm to find largest planar section of a convex polyhedral solid
We add a bit more on shadows and planar sections following On a pair of solids with both corresponding maximal planar sections and shadows having equal area . We consider only polyhedrons.
Given a ...
- 5,979
3
votes
0
answers
136
views
If all max area planar sections of a solid are centrally symmetric, will the solid as whole be centrally symmetric?
It is known that every planar section of an ellipsoid is an ellipse - a centrally symmetric planar figure.
Are there convex solids other than ellipsoids with the property that all its planar sections ...
- 5,979