Questions tagged [simplex]

For questions on the $n$-simplex, an $n$-dimensional polytope with $n+1$ points.

A [simplex][1] is a higher dimensional analogue of a triangle or tetrahedron. The number of faces in a simplex can be determined using Pascal's triangle.

[1]: https://en.wikipedia.org/wiki/Simplex#:~:text=In%20geometry%2C%20a%20simplex%20(plural,0%2Dsimplex%20is%20a%20point%2C&text=a%204%2Dsimplex%20is%20a%205%2Dcell.

759 questions
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PDF of volume of tetrahedron with random coordinates

Question What is the probability distribution function (PDF) of the absolute volume of a tetrahedron with random coordinates? The 4 random tetrahedron vertices in $\mathbb{R}^3$ are $$ \mathbf{\mathrm{X}_1} =(x_1^1,x_1^2,x_1^3),\;\; …
15
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Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$ \sum_{i=1}^N x_i = 1 $$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform distribution $[0,1]$ and then I transform them into $x_i$…
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Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$

Let $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$. I know $T_n$ is tetrahedron. My question: How can I compute the volume of $T_n$ for every $n$?
user145801
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Definition of simplex

From Wikipedia: an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. I was wondering if the definition is equivalent to say a simplex is synonym of a convex polytope? Is simplex defined only for…
Tim
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Question about identifying pairs of edges of disjoint $2$ simplices

This exercise $2.1.10$ in page $131$ of Hatcher's book Algebraic topology. (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface, locally homeomorphic to…
Anubhav Mukherjee
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Volume of the intersection of two simplexes

Let $S_n$ be the interior of the unitary $n$-simplex, i.e $ S_n =\{{\bf x} \in \mathbb{R}^n \mid x_i\ge0 \wedge \sum_{i=1}^n x_i\le1\}$ Let $T_n({\bf y})$ be the reversed simplex with origin at ${\bf y}$, ie $T_n({\bf y}) = \{{\bf x} \in…
leonbloy
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Volume of an n-simplex

It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$. Is there an easier way to see this?
MathPhys137
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Expected tetrahedron volume from normal distribution

Two equivalent formulas for the volume of a random tetrahedron are given. Further on you can find an interesting conjecture for the expected volume that shall be proved. Tetrahedron volume Given are 12 independent standard normal distributed…
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Intuition for volume of a simplex being 1/n!

Consider the simplex determined by the origin, and $n$ unit basis vectors. The volume of this simplex is $\frac{1}{n!}$, but I am intuitively struggling to see why. I have seen proofs for this and am convinced, but I can't help but think there must…
doctorpigeonhole
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More approximately orthogonal vectors than the dimension of the space

It is impossible to find $n+1$ mutually orthogonal unit vectors in $\mathbb{R}^n$. However, a simple geometric argument shows that the central angle between any two legs of a simplex goes as $\theta = \mathrm{arccos}(-1/n)$. This approaches $90$…
Nick Alger
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The difference between an affine k-simplex and a rectilinear k-simplex

The notion of rectilinear k-simplex appears in Theorem 10.27 of Rudin's book "Principles of Mathematical analysis", then what is the definition of a rectilinear k-simplex? I read the proof of Theorem 10.27 and think that the proof treats oriented…
nick
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PDF of area of triangle with normally-distributed coordinates in any dimensions

Question What is the probability distribution function (PDF) of the absolute area of a triangle with normally-distributed coordinates in $\mathbb{R}^m$ $(m \in \mathbb{N}, m\ge2)$ ? A conjecture is given that can be proved or might help to find the…
8
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2 answers

PDF of distance from origin to simplex in high dimension

Let $\Delta^{n-1}$ be the standard simplex in $n$ dimensions: $$ \Delta^{n-1} = \{ \mathbf{x}\in \mathbb{R}^n: \sum_i x_i=1 , \mathbf{x}\geq0\} $$ And assuming that we are uniformly sampling points $\mathbf{X}$ from this simplex, then the Euclidean…
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Can a regular $n$-simplex have vertices in $\mathbb Z^n$ for $n > 1$?

Trivially, a regular $0$-simplex (point) and $1$-simplex (line segment) can have integer vertices in $0$ and $1$ dimensional Euclidean space respectively. On the other hand, a regular $2$-simplex (equilateral triangle) cannot have vertices in…
Peter Kagey
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What is the difference between a unit simplex and a probability simplex?

The unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, \ldots,e_n\in\mathbf R^n$. It can be expressed as the set of vectors that satisfy $$x\succcurlyeq0,\quad\mathbf 1^\mathrm T…
W. Zhu
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