Questions tagged [simplicial-volume]

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An alternative to Cayley Menger determinant for calculating simplex volume

I recently came across the determinant of a symmetric $3\times 3$ matrix $\begin{pmatrix} 2a^2& a^2+b^2-c^2& a^2+d^2-e^2\\ a^2+b^2-c^2& 2b^2& b^2+d^2-f^2\\ a^2+d^2-...
Manfred Weis's user avatar
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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 ...
van der Wolf's user avatar
1 vote
0 answers
151 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 ...
Ben Webster's user avatar
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3 votes
1 answer
184 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’,...
Manfred Weis's user avatar
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2 votes
1 answer
134 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 ...
Bodo Manthey's user avatar
7 votes
2 answers
816 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}^...
Erik's user avatar
  • 81
1 vote
1 answer
128 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 ...
Shivin Srivastava's user avatar
5 votes
1 answer
168 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 ...
Bálint Czúcz's user avatar
1 vote
0 answers
89 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 \...
Frank Hellmann's user avatar
2 votes
0 answers
248 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 ...
cata's user avatar
  • 337
1 vote
1 answer
553 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 ...
edgar's user avatar
  • 35
3 votes
1 answer
466 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 ...
Dima Pasechnik's user avatar
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 ...
John Pardon's user avatar
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9 votes
1 answer
710 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 ...
John Pardon's user avatar
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4 votes
2 answers
3k 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 ...
mr grumpy's user avatar
  • 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 ...
cdouglas's user avatar
  • 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 ...
Danny Calegari's user avatar