Questions tagged [simplicial-volume]
The simplicial-volume tag has no usage guidance.
16
questions
1
vote
0
answers
82
views
If all the chocolate is within distance r of the outer boundary of the choco egg, what is the max.quantity of chocolate contained within a unit ball?
We have proved the following statement, but wonder if this result is actually known (reference??)
It solves the following problem. Suppose you have a (possibly) hollow chocolate egg whose outer ...
- 151
1
vote
0
answers
139
views
Volume of a polytope as its degenerates to be lower dimensional
Consider a polytope $P$ defined by the usual inequalities $A\mathbf{x}\leq \mathbf{b}$; let me assume that $P$ is not contained in a proper subspace. A result which I believe to true, but am not ...
- 42.9k
3
votes
1
answer
171
views
Deflating a tetrahedron to a $K_4$ graph with equal changes to sidelengths
Let $A,B,C,D$ be the corners of a tetrahedron with positive volume and distinct sidelengths. Is there a positive $x$ and a planar straight-line embedding of a $K_4$ graph with distinct vertices $A’,B’,...
- 12k
2
votes
1
answer
124
views
Maximal area/volume of (d-1)-dimensional object in d-dimensional hypercube
What is the largest area/volume of any $(d-1)$-dimensional flat object that fits into the $d$-dimensional unit hypercube? For instance, for $d=2$, the answer is $\sqrt 2$, as this is the length of the ...
- 177
7
votes
2
answers
667
views
Formula for volume of a convex polytope
So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
- 81
1
vote
1
answer
121
views
Number of distinct points in an n-dimensional tetrahedron
Consider an n-dimensional tetrahedron with $n+1$ vertices $\langle v_0, v_1, \dots,v_n\rangle$. $v_0$ is the origin while $v_i$ lies on $e_i$ (the $i^{th}$ coordinate axis) at a distance $D$ from the ...
5
votes
1
answer
157
views
Volume satisfying inequality constraints (simplex subset)
Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?
$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le ...
- 51
1
vote
0
answers
83
views
Volume under the intersection of scaled simplices
This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space:
$\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 \...
2
votes
0
answers
246
views
Volume of bounded regions in hyperplane arrangements
I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...
- 337
1
vote
1
answer
541
views
Hypervolume of n-d simplex in an n+1 space
Hello,
This is my first time asking a question on this site so please let me know if I'm doing it wrong.
I have been trying to find out how to compute the hypervolume of an n-d simplex in an n+1 ...
- 35
3
votes
1
answer
449
views
from affine matroid to measures
Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the ...
- 13.3k
20
votes
1
answer
1k
views
Why is there a unique hyperbolic simplex of largest area?
Why is there a unqiue ideal $n$-simplex in $\mathbb H^n$ with largest volume for $n\geq 3$?
For $n=3$, this is a standard calculation, and for larger dimensions is much harder (see Haagerup and ...
- 17.9k
9
votes
1
answer
692
views
Is there a simplicial volume definition of Chern Simons invariants?
Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...
- 17.9k
4
votes
2
answers
2k
views
Computational geometry, tetrahedron signed volume
Short version: I'm trying to compute the orientation of a triangle on a plane, formed by the intersection of 3 edges, without explicitly computing the intersection points.
Long version: I need to ...
- 143
20
votes
4
answers
2k
views
Is there a volume conjecture for closed 3-manifolds?
A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is
Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...
- 3,083
9
votes
5
answers
1k
views
Simplicial volume
Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm")?
(Added:) The ...
- 2,532