All Questions
1,544 questions
8
votes
0
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118
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Defining convex sums locally on the sphere?
$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...
- 637
1
vote
0
answers
68
views
Generators of $\operatorname{NE}(X)$
Let $X$ be a smooth projective variety and $\operatorname{Nef}(X)$ is a rational polyhedron. This implies that the dual $\overline{\operatorname{NE}}(X)$ is a rational polyhedron as well. Suppose we ...
- 335
0
votes
0
answers
115
views
Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
- 1
0
votes
0
answers
72
views
Probability of being inside a convex n-dimensional polytop
I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays.
As part of the research, each individual transmission is modelled using a probability ...
1
vote
0
answers
42
views
On a pair of solids with both corresponding maximal planar sections and shadows having equal area
This post pulls together Are two convex solids with all corresponding shadows equal in area congruent? and
What can be said about 2 convex solids with corresponding maximal planar sections having ...
- 5,979
1
vote
0
answers
59
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What can be said about 2 convex solids with corresponding maximal planar sections having equal area?
This post follows Are two convex solids with all corresponding shadows equal in area congruent?
Every convex 3D body has planar sections with normals in any given direction. We consider the maximum ...
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2
votes
1
answer
302
views
Are two convex solids with all corresponding shadows equal in area congruent?
By shadow we mean the orthogonal projection of a convex 3D body P onto a 2D plane, for example, the shadow on the xy-plane, with P above (z>0) that plane and the light at L=(0,0,+∞). P an be freely ...
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1
vote
0
answers
79
views
Touring a sequence of convex polygons with minimal energy
There is a known problem of touring a sequence of $n$ polygons: given a starting point $s$, an ending point $t$ and a sequence of polygons $P_1,\dots,P_k$ with a total of $n$ vertices, find points $...
- 177
5
votes
0
answers
145
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Are there convex polyhedrons that can be cut into mutually congruent connected pieces only if pieces are non-convex?
This is the 3D (and higher D) version of A claim on partitioning a convex planar region into congruent pieces
Is there a 3D convex polyhedral solid that can be cut into 2 mutually congruent non-...
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2
votes
0
answers
51
views
Convex polygons that can be cut into sets of m mutually congruent convex pieces in exactly n ways
General question: Given two integers m and n, to find a convex polygonal region that can be cut into sets of m mutually congruent convex pieces in exactly n ways - the shape of pieces in each set ...
- 5,979
5
votes
1
answer
176
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Efficient counting of integer solutions to linear system
In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
- 151
2
votes
0
answers
62
views
On convex polygons that can be cut into convex and mutually congruent pieces in exactly one way
Observations: any thin isosceles triangle has exactly 1 partition into 2 congruent pieces - only 1 line, bisector of its apex, does it.
By attaching a right triangle with base 1 and altitude 2 to an ...
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6
votes
2
answers
404
views
Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
- 307
1
vote
0
answers
93
views
Inside-out dissections of a cube
Ref:
Inside-out polygonal dissections
Inside-out dissections of solids
Definitions: A polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P ...
- 5,979
4
votes
2
answers
154
views
Pushing a convex cone and equidistants
Let $K$ be a closed convex cone in an n-dimensional Euclidean space.
Suppose $K$ has non-empty interior. For $t > 0$
form the subcone $K_t$ consisting of all points in $K$ which lie a ...
- 8,336
1
vote
0
answers
50
views
Interpolation in convex hull
I'm reading a paper, Learning in High Dimension Always Amounts to Extrapolation, that provides a result I don't understand.
It provides this theorem which I do understand:
Theorem 1: (Bárány and ...
6
votes
0
answers
48
views
Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?
The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
- 3,068
1
vote
1
answer
98
views
On centrally symmetric convex figures on the hyperbolic plane
A planar region C such that there is an interior point that bisects all chords of C that passes through it may be termed centrally symmetric. It appears that such figures exist in non-Euclidean ...
- 5,979
0
votes
1
answer
91
views
On nontrapping manifolds
Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary.
Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is ...
- 4,135
0
votes
0
answers
42
views
On cutting polyhedrons into convex polyhedral pieces all with same volume, surface area and total edge length
This is a constrained version of the 'fair partition' ('spicy chicken' - https://arxiv.org/abs/1306.2741) question.
It seems that there are convex polyhedrons that cannot be cut into n convex pieces ...
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0
votes
1
answer
316
views
A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 153
1
vote
0
answers
44
views
On area bisectors and perimeter bisectors of planar convex regions
We try to proceed from A claim on the concurrency of area bisectors of planar convex regions
Definitions: Given a planar convex region C, an area bisector of C is any line segment that partitions C ...
- 5,979
1
vote
2
answers
125
views
Existence of a short path in a convex graph drawing
Consider a (simple) convex drawing of a $3$-connected planar graph $G$. (Edit:) Let us also assume that the border polygon has at least $4$ sides.
My question is as follows: does there exist a simple ...
- 307
1
vote
0
answers
40
views
Why disks might be special - on chords that cut off segments of a specified area from a planar convex body
This post presents a variant of On segments of equal area cut from planar convex regions by chords
Consider a planar convex region C of unit area and all chords of it that cut off a segment of area α ...
- 5,979
1
vote
1
answer
94
views
Why are symmetric convex bodies with a smooth boundary and non-vanishing Gaussian curvature of particular interest in harmonic analysis?
I don't work in harmonic analysis or convex analysis, but in some literature of harmonic analysis, I often see the assumption that "let $K$ be a symmetric convex body with a smooth boundary and ...
- 457
1
vote
0
answers
86
views
Busemann-Petty type problems on complex vector spaces [closed]
We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...
0
votes
0
answers
40
views
Iterating partially-unconstrained optimization with projection
Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize
$$
\tag{(A)...
- 5,405
1
vote
0
answers
95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
1
vote
0
answers
91
views
Does convexity of boundary implies geodesic convexity?
I came across the following result (mentioned on Pg. 3 of this talk) that states that
If $D$ is an open connected subset of a complete Riemannian manifold with smooth metric then $\partial D$ convex ...
- 537
2
votes
1
answer
197
views
Prékopa-Leindler style inequality?
Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
- 125
0
votes
0
answers
27
views
'$\alpha$-moments' and '$\alpha$-centers' of planar convex regions
We try to proceed from Least area and least perimeter triangles that contain a convex planar region - how different can they be?
The partial answer given to the above question shows a convex ...
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6
votes
1
answer
413
views
How many unit cubes are needed to 'hide' a unit cube fully in 3D?
Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
- 5,979
2
votes
2
answers
201
views
On special points within convex solids with all planar sections passing through them having equal area
Question: If within a convex solid body C there is a special point P such that every planar section of C passing through P has the same area, then, can we assert that C is a sphere and P its center? ...
- 5,979
1
vote
0
answers
32
views
Variants of cutting plane method for convex optimization
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
- 139
0
votes
0
answers
32
views
Set of enclosed convex polyhedra in a graph
Given a straight-line graph embedded in $\mathbb{R}^3$ with known vertex coordinates and edges and no edge intersections, is it possible to find all the enclosed convex polyhedra inside? If so, is ...
- 1
1
vote
1
answer
84
views
Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
0
votes
0
answers
22
views
Directions of differentiability of log-concave measures with infinite-dimensional support
I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
- 651
0
votes
1
answer
133
views
Centroid of Minkowski sum
Let $A$ and $B$ be two compact convex subsets of $\mathbb{R}^n, n\geq 2$. Assume $x_A$ and $x_B$ are their respective centroid. If we form the Minkowski sum $C=A+B = \{x+y\mid x\in A, y\in B\}$, what ...
- 161
4
votes
0
answers
46
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Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
0
votes
1
answer
103
views
Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
- 857
1
vote
0
answers
40
views
Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
- 5,979
1
vote
0
answers
96
views
Polyhedrons with mutually non-congruent faces, all of equal area
This question is closely related to Convex polyhedra with non-congruent faces
It is known that if all faces of a tetrahedron ought to have same area (or same perimeter), then, the faces are ...
- 5,979
1
vote
1
answer
205
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Van der Waerden conjecture and Alexandrov-Fenchel inequality
The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel ...
- 21.8k
7
votes
3
answers
703
views
A continuous version of Carathéodory's convex hull theorem
A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
- 7,149
0
votes
0
answers
39
views
Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101
0
votes
0
answers
30
views
Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
- 23
1
vote
1
answer
98
views
To place copies of a planar convex region such that number of 'contacts' among them is maximized
A contact between two planar convex regions obviously happens either along a line segment or at a single point.
Question: Given a planar convex region $C$ and a number $n$, we need to lay out $n$ ...
- 5,979
4
votes
1
answer
197
views
What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?
Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
- 32.5k
0
votes
0
answers
49
views
Which planar convex region with specified area and perimeter maximizes/minimizes Moment of Inertia?
By moment of inertia of a planar convex region C, here we mean its moment of inertia about an axis passing through the center of mass of C and perpendicular to the plane of C.
Question: For specified ...
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8
votes
1
answer
473
views
Is the gradient of a strictly convex, continuously differentiable function a homeomorphism?
Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $f:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.
$\nabla f$ is a bijection, but is ...
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