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# Questions tagged [simplicial-volume]

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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 ...
1 vote
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...
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 ...