N维单形的体积

青青子衿

七，$n$ 维单形不等式
$\mathbb  R^n$ 中子集 $E$ 的任两点连线仍在 $E$ 中，称 $E$ 为凸体，包含 $R^*$ 中 $n_{-}$点 $A_1, \cdots, A_{n+1}$ 的最小凸体，称为由 $\left\{A_1, \cdots, A_{n+1}\right\}$ 张成的 $n$ 维单形 $n$-simplex $\sum(A)$ ．若 $A_1$ 的坐标为 $\left(x_{i 1}, \cdots, x_k\right)(1 \leqslant k \leqslant n+1)$ ，则 $\sum(A)$ 的体积 $V(A)$为
\[
V(A)=\frac{1}{n!}\left|\begin{array}{ccccc}
x_{11} & x_{12} & \cdots & x_{12} & 1 \\
x_{11} & x_{22} & \cdots & x_{2 n} & 1 \\
\cdots & & & & \\
x_{k+1,1} & x_{n+1,2} & \cdots & x_{n+1, *}
\end{array}\right|
\]
当 $n \neq 0$ 时，$n$ 维单形有两个定向，用頂点的顺序给出．注意彼此相差一个偶排列的利，序代表同一个定向．
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 $\Delta^0$.

$\Delta^0$	$\Delta^1$	$\Delta^2$	$\Delta^3$

hbghlyj

\begin{array}{|l|l|}
\hline \text { circumradius } & \sqrt{\frac{n}{2 n+2}} s \\
\hline \text { edge midradius } & \frac{1}{2} \sqrt{\frac{n-1}{n+1}} s \approx 0.5 \sqrt{\frac{n-1}{n+1}} s \\
\hline \text { face midradius } & \frac{\sqrt{\frac{n-2}{n+1}} s}{\sqrt{6}} \approx 0.408248 \sqrt{\frac{n-2}{n+1}} s \\
\hline \text { inradius } & \sqrt{\frac{1}{2 n^2+2 n}} s \\
\hline \text { content } & \frac{\sqrt{2^{-n}(n+1)} s^n}{n!} \\
\hline \text { hyper-surface area } & \frac{\sqrt{2^{1-n} n}(n+1) s^{n-1}}{(n-1)!} \\
\hline
\end{array}
(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

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