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Mathematics Stack Exchange

Questions tagged [simplex]

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

A Simplex is a higher dimensional analogue of a triangle or tetrahedron. It represents the simplest possible polytope made with line segments in any given dimension. The number of faces in a simplex can be determined using Pascal's triangle.

Below is an image of Simplex's of the first, second, third, and fourth dimensions from Wikipedia:

enter image description here

Reference:

841 questions
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Intuition for volume of a simplex being $\frac 1{n!}$

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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18
votes
2 answers

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),\;\; …
granular_bastard
  • 563
17
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4 answers

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
17
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2 answers

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 $x_i$…
LordM'zn
  • 171
16
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3 answers

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
  • 6,708
14
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4 answers

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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13
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1 answer

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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12
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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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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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11
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1 answer

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…
granular_bastard
  • 563
10
votes
1 answer

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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10
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1 answer

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
  • 631
10
votes
1 answer

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…
granular_bastard
  • 563
9
votes
0 answers

Proof of the general case of Feynman's integration trick

I want to show that $$\frac{1}{\displaystyle\prod_{i=0}^{i=n}A_{i}}=n!\int\limits_{|\Delta^{n}|}\frac{\mathrm d\sigma}{\left( \displaystyle \sum\limits_i s_i A_i \right)^n}$$ where $\mathrm d\sigma$ is the Lebesgue measure on the standard…
dbrane
  • 241
9
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Geometric interpretation of duality in optimization

There are several beautifully written posts on stackexchange about duality. For example: A technical explanation of duality that attempts to offer some intuitions including the insight that the primal and dual each sort of encode the other, so that…
John Strong
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