Question

# 问题

Questions in category: 度量几何 (Metric Geometry).

## $\inf_{f\in\alpha}\mathrm{dil}(f)$ 为代表 $\alpha$ 的最短闭曲线 $\gamma_0$ 的长度.

Posted by haifeng on 2020-07-05 23:45:34 last update 2020-07-06 22:18:23 | Answers (2) | 收藏

$\mathrm{dil}(f)=\sup_{t\neq t'\\ t,t'\in I}\frac{d(f(t),f(t'))}{d(t,t')}$

$f(t)=\begin{cases} t, & t\in[0,1]\\ 2t-1, & t\in(1,2]\\ 3t-3, & t\in(2,3] \end{cases}$

$f(t)=\begin{cases} e^{i\frac{\pi}{2}t}, & t\in[0,1]\\ e^{i(\pi t-\frac{\pi}{2})}, & t\in(1,2]\\ e^{i\frac{\pi}{2}(t+1)}, & t\in(2,3] \end{cases}$

$\mathrm{dil}(f)=\sup_{t\neq t'\\ t,t'\in I}\frac{d(f(t),f(t'))}{d(t,t')}=\pi$

$h(s)=\begin{cases} e^{i\frac{\pi}{2}3s}, & s\in[0,\frac{1}{3}]\\ e^{i(\pi 3s-\frac{\pi}{2})}, & s\in(\frac{1}{3},\frac{2}{3}]\\ e^{i\frac{\pi}{2}(3s+1)}, & t\in(\frac{2}{3},1] \end{cases}$

$\mathrm{dil}(h)=\sup_{s\neq s'\\ s,s'\in I}\frac{d(h(s),h(s'))}{d(s,s')}=3\pi$

$\mathrm{dil}(h)=\mathrm{dil}(f)\cdot\mathrm{dil}(g).$

$\|\alpha\|=\inf_{f\in\alpha}\mathrm{dil}(f)\cdot\mathrm{vol}(B^1)$

$\mathrm{length}(\gamma)=\int_I |\dot{\gamma}|ds$

$\pi_n(X,x_0)$ 上的群结构