Questions tagged [convex-polytopes]
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
942 questions
5
votes
2
answers
368
views
Diameter-area ratio for affine tranformations.
Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...
- 5,063
3
votes
0
answers
148
views
Is the chromatic number of the graph of the permutahedron known?
The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the ...
- 101
3
votes
1
answer
518
views
n-dimensional Delaunay Triangulation of Lattices
I have several questions concerning the Delaunay triangulation of a high dimensional lattice.
Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
- 31
6
votes
1
answer
715
views
Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
- 1,318
3
votes
2
answers
270
views
Formalization (and background) of a formula, concering the integral points of a polygon.
I have recently become aware of the following neat statement.
Consider a convex polygon $P$ in the real plane with integral vertices. If we associate with every integral point $(a,b)$ the monomial $x^...
- 3,513
3
votes
2
answers
115
views
algorithm of polytope
Basically, we have an incremental sets of vertices
$${V_1} \subset {V_2} \subset ...$$
for each of them, we could build a polytope $${P_i} = Conv({V_i})$$
Consequently, we can compute
$${F_i} = ...
- 867
3
votes
1
answer
294
views
convex hull of the set of permutations with one cycle
is there a way to describe the convex hull of the set of permutation matrices with exactly one non-trivial cycle?
or maybe I should ask for the convex hull of cycle matrices :
let $(i_{1},..,i_{k})$ ...
- 33
1
vote
0
answers
168
views
Projecting on a a special polyhedron
Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$
Notice that $\mathcal P_X$ is symmetric about the origin.
...
- 6,853
0
votes
0
answers
40
views
Influence of Vertex Weights on Performance of Polyhedral TSP Algorithms
It is well known, that the optimal tours are invariant under the addition of vertex weights; it is also known, that minimum spanning trees provide an upper bound on the length of the shortest Hamilton ...
- 13.2k
8
votes
0
answers
826
views
Geometry of the metric cone
Let us say that two metrics $d$ and $d_0$ on a set $X$ are related if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
...
- 13.5k
2
votes
0
answers
62
views
on vectors for which the intersection of their convex hull and the nonegative orthant is the unit simplex
Consider the vectors $r^1 = (0,2,-1)$, $r^2 = (-1,0,2)$, and $r^3 = (2,-1,0)$. Two properties of these vectors that interest us here are:
1) The $i$'th coordinate of $r^i$ is 0, and
2) The ...
- 745
4
votes
0
answers
86
views
Polynomial approximations on the Boolean hypercube
Given $\vec{a} \in \mathbb{R}^n$ and $b \in \mathbb{R}$ consider the function $f(x) = Th[\vec{a}.\vec{x}+b]$ on $x \in \{-1,1\}^n$ such that the ``threshold function (Th)" gives $1$ when the argument ...
- 2,246
5
votes
0
answers
309
views
Which (polytopal) fans/polytopes are secondary?
Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...
- 902
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 3
4
votes
0
answers
158
views
Reference for the notion of polyhedra "degenerations"
Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ (...
- 3,513
6
votes
0
answers
118
views
Rational $d$-simplices
Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and ...
- 151k
3
votes
1
answer
140
views
convex decompositions of the sphere
Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...
- 96.4k
4
votes
2
answers
1k
views
Finding smallest ellipsoid that circumscribes over intersection of two ellipsoids that do not have common center
Does this already exist in literature? The closest Ive been able to find is circumscribe intersection of two ellipsoids with a common center by W. Kahan (http://www.cs.berkeley.edu/~wkahan/Ellipint....
- 263
2
votes
2
answers
91
views
Can we prove that a normal surface of an extreme point of a convex subset of a simplex is a separating hyperplane?
Let's assume that we have a simplex $G = \{x\in R^d|\sum_{i=1}^d x_i=1, x_i\ge 0 , i = 1, 2, .., d\}$ and a polyhedral convex subset $H \subseteq G$.
Is it possible to prove that for any extremal ...
2
votes
0
answers
210
views
Finding optimal linear transformation for intersection of convex polytopes
I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) ...
- 695
2
votes
1
answer
141
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 ...
- 21
1
vote
1
answer
284
views
Linear map of Zonotopes [closed]
Consider a linear system with a map $A: \mathbb{R}^n \rightarrow
\mathbb{R}^m$ such that $y = A x$ with $n \geq m$
The input space $x$ is constrained by a zonotope set $\mathcal{X}
\subseteq \mathbb{...
4
votes
1
answer
175
views
Are all marked order polytopes normal?
Richard Stanley showed that order polytopes have a unimoudlar triangulation.
In particular, this implies that they are integrally closed/normal.
One can generalize order polytopes to marked order ...
- 15.8k
19
votes
0
answers
577
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 ...
- 151k
3
votes
0
answers
527
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 ...
- 10.5k
8
votes
1
answer
2k
views
About the surface area vs. volume of polytopes
Given a convex body $K\in\mathbb{R}^n$, represented by a set of linear inequalities (intersection of halfspaces), I am interested in understanding how much of its volume can be close to its perimeter (...
- 81
4
votes
0
answers
282
views
Prove or disprove a claim about covering a polytope by convex polytopes in a certain way
Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes $\{K_i\}_{...
- 1,350
1
vote
2
answers
129
views
An optimality condition for the corners of convex polytopes?
Let $H,H'\subset\mathbb{R}^m$ be two hyperplanes with unit normal-vector,
and let $P\subset\mathbb{R^m}$ be a convex polytope (defined via its corners $v_0, ... , v_n$, where $n\ge m$).
Let's ...
- 13.2k
1
vote
0
answers
62
views
Sampling in a polyhedral complex
Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...
- 2,246
8
votes
2
answers
385
views
Volumes of convex vs non-convex polyhedra with prescribed facets areas
It is known that given a set of Areas $A_f$ and normals $\vec{n}_f$ if $\sum_f A_f \vec{n}_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov ...
- 83
2
votes
2
answers
1k
views
Convex upper bound on a linear-fractional function
I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ...
- 362
0
votes
1
answer
124
views
polynomial expression for counting number of integral points of a set
Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...
- 1,713
5
votes
0
answers
136
views
Face structures of chain polytopes
For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<...
- 3,513
3
votes
1
answer
274
views
Criteria for abstract polytopes to be convex polytope
Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
- 2,023
2
votes
0
answers
214
views
About optimizing a convex function on a hypercube
Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...
14
votes
0
answers
262
views
Polytopes with few vertices and few facets
I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...
- 278
1
vote
1
answer
337
views
Estimating the volume of a union of balls
Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where $\...
user24451
12
votes
1
answer
383
views
Connecting Lemma in the Alexandrov's existence theorem.
At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem.
Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral
metrics on the $\mathbb S^...
- 45k
4
votes
1
answer
333
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
- 195
4
votes
1
answer
510
views
moduli space of polytopes
When considering classification problems about polytopes, I sometimes has the feeling that one need to talk about certain parametrized families, i.e. moduli space of such polytopes. But neither do I ...
- 3,472
2
votes
1
answer
2k
views
Proving that a specific function is quasiconvex
Hello all,
Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function $f_{...
- 43
2
votes
0
answers
351
views
Looking for the manuscript "Uniform polytopes" by N. Johnson
The manuscript Uniform Polytopes (1991) by Norman Johnson is cited in the wikipedia page on uniform polytopes (http://en.wikipedia.org/wiki/Uniform_polytope).
Is there an electronic copy of this ...
- 838
0
votes
0
answers
167
views
Centrally symmetric neighborly polytopes
Let $P$ be a centrally symmetric $k$-neighborly $d$-polytope. Let $ P^* $ is polar (also dual) of $P$. Consider a polytope $Q^*$ , $Q^* \subset P^* $(not necessarily a face of $P^*$). By duality,
$P\...
- 31
2
votes
2
answers
459
views
Bound on the (anticanonical) degree of toric Fano varieties
Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K_X)^n$ ...
- 13.5k
4
votes
1
answer
1k
views
Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability)
Assume I have polytope in R^k given by N (k<< N) linear inequalities (A_i x < b_i).
I guess complexity of its volume calculate is higher than linear in "N", am I right ?
(Is the complexity ...
- 24.9k
0
votes
1
answer
102
views
Mapping n variables to the surface of an n+1-dimensional cross polytope [closed]
I'm looking for a function that maps n variables to points on the surface of an n+1-dimensional cross polytope. For example, given one variable, the function would return a point on the perimeter of ...
0
votes
3
answers
533
views
How to determine if two rational cones intersect?
Let $$\displaystyle C_1=C(r_1,...,r_{n_1})=( x\in \mathbb{R^d}|x=\sum_{i=1}^{n_1}\lambda_i r_i, \lambda_i\in \mathbb{R_{>0}})$$
$$\displaystyle C_2=C(t_1,...,t_{n_2})=(x\in \mathbb{R^d}|x=\sum_{i=...
- 360
3
votes
1
answer
233
views
Equiprojective polyhedra
Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open,
and which some might find intriguing.
Define an ...
- 151k
13
votes
0
answers
406
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?
- 131
2
votes
0
answers
253
views
Newton polyhedron and product of ideals
Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and
$J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ,\...
- 1,713