N维单形的体积

青青子衿

怎么证明？？

icesheep

标准单纯形+仿射变换

hbghlyj

Topology
Definition 4.2. The standard $n$-simplex[标准单纯形] is the set
$$
\Delta^n=\left\{\left(x_1, \ldots, x_{n+1}\right) \in \mathbb{R}^{n+1}: x_i \geq 0 \forall i \text { and } \sum_i x_i=1\right\} .
$$
The non-negative integer $n$ is the dimension of this simplex. Its vertices, denoted $V\left(\Delta^n\right)$, are those points $\left(x_1, \ldots, x_{n+1}\right)$ in $\Delta^n$ where $x_i=1$ for some $i$ (and hence $x_j=0$ for all $j \neq i$ ). For each non-empty subset $A$ of $\{1, \ldots, n+1\}$ there is a corresponding face of $\Delta^n$, which is
$$
\left\{\left(x_1, \ldots, x_{n+1}\right) \in \Delta^n: x_i=0 \forall i \notin A\right\} .
$$
Note that $\Delta^n$ is a face of itself (setting $A=\{1, \ldots, n+1\}$ ). The inside of $\Delta^n$ is
$$
\operatorname{inside}\left(\Delta^n\right)=\left\{\left(x_1, \ldots, x_{n+1}\right) \in \Delta^n: x_i>0 \forall i\right\}
$$
Note that the inside of $\Delta^0$ is $\Delta^0$.

$\Delta^0$	$\Delta^1$	$\Delta^2$	$\Delta^3$
		import three;import graph3; unitsize(4cm);settings.render=2; triple A, B, C; currentprojection=perspective(1,2,1); A = (1,0,0); B = (0,1,0); C = (0,0,1); draw(surface(A--B--C--cycle),lightyellow+opacity(.8),nolight); draw(A--B--C--cycle); label("$A$", A, -Y); label("$B$", B, -X); label("$C$", C, -X-Y); xaxis3(Label("$x$",EndPoint),Arrow3(),xmax=1.4); yaxis3(Label("$y$",EndPoint),Arrow3(),ymax=1.4); zaxis3(Label("$z$",EndPoint),Arrow3(),zmax=1.4); import three;import graph3; unitsize(4cm);settings.render=2; triple A, B, C; currentprojection=perspective(-1,2,1); A = (1,0,0); B = (0,1,0); C = (0,0,1); draw(surface(A--B--C--cycle),lightyellow+opacity(.8),nolight); draw(A--B--C--cycle); label("$A$", A, W); label("$B$", B, E); label("$C$", C, N); xaxis3(Label("$x$",EndPoint),Arrow3(),xmax=1.4); yaxis3(Label("$y$",EndPoint),Arrow3(),ymax=1.4); zaxis3(Label("$z$",EndPoint),Arrow3(),zmax=1.4);	import three; unitsize(4cm);settings.render=2; triple A, B, C, D; currentprojection=perspective(0,-5,3); A = (1,0,1); B = (0,1,1); C = (0,0,0); D = (1,1,0); draw(A--B--C--cycle); draw(A--D); draw(B--D); draw(C--D); label("$A$", A, E); label("$B$", B, NW); label("$C$", C, SE); label("$D$", D, S); draw(surface(A--B--C--cycle),opacity(.4)+lightmagenta); draw(surface(A--D--B--cycle),opacity(.4)+lightmagenta); draw(surface(D--C--B--cycle),opacity(.4)+lightmagenta); draw(surface(A--C--D--cycle),opacity(.4)+lightmagenta);

hbghlyj

Simplex[{p1,…,pk}] represents the simplex spanned by points pi.
Simplex is also known as point, line segment, triangle, tetrahedron, pentachoron, hexateron, etc.
Simplex represents all convex combinations of the given points $\Set{\lambda_{1} p_{1}+\cdots+\lambda_{k} p_{k} \mid \lambda_{i} \geq 0 \wedge \lambda_{1}+\cdots+\lambda_{k}=1}$. The region is $k-1$ dimensional when $p_i\in\Bbb R^n$ are affinely independent and $n\ge k-1$.
Example simplices where rows correspond to embedding dimension:

Simplex[n] for integer $n$ is equivalent to Simplex[{{0,…,0},{1,0,…,0},…,{0,…,0,1}}], the unit standard simplex in $\Bbb R^n$.

hbghlyj

WolframAlpha

（assuming embedding dimension $n$ and edge lengths $s$）

hbghlyj

hbghlyj 发表于 2023-2-25 10:21
$p_i\in\Bbb R^n(i=1,\cdots,n+1)$ are affinely independent

对应于1#的行列式 $\left|\begin{array}{ccccc}x_{11} & x_{12} & \cdots & x_{1 n} & 1 \\ x_{21} & x_{22} & \cdots & x_{2 n} & 1 \\ \cdots &\cdots&\cdots&\cdots&\cdots\\ x_{n+1,1} & x_{n+1,2} & \cdots & x_{n+1, n} & 1\end{array}\right|\ne0$

hbghlyj

Asymptote画一个正四面体棱切球
import three;
unitsize(2cm);
currentprojection=perspective(1,1,2);
// Define the vertices of the tetrahedron
triple A = (1, 1, 1);
triple B = (-1, -1, 1);
triple C = (-1, 1, -1);
triple D = (1, -1, -1);

// Define the center of the sphere
triple O = (0,0,0);

// Draw the sphere
draw(unitsphere, white+opacity(.2));

// Draw the tetrahedron
defaultpen(lightblue);
draw(surface(A--B--C--cycle));
draw(surface(A--B--D--cycle));
draw(surface(A--C--D--cycle));
draw(surface(B--C--D--cycle));

lihpb

1楼是什么书

青青子衿

lihpb 发表于 2024-9-7 16:53
1楼是什么书

“常用不等式”  匡继昌