Two little remarks on sphere

1. "Take a sphere of radius $R$ in $N$ dimensions, $N$ large; then most points inside the sphere are in fact very close to the surface." David Ruelle

Let $R>0$ and $0<\alpha <1$. Let $$B(x;R)=\{(x_1,\cdots ,x_N):x_1^2+\cdots +x_N^2\le R^2\}.$$ The fraction of the volume of $B(x;\alpha R)$ to the volume of $B(x;R)$ is ${\alpha }^N$, and $\underset{N\to 0}{lim}{\alpha }^N=0$. It means that "a full sphere of high dimension has all its volume within a "skin" of size $\epsilon$ near the surface" (Collet and Eckmann). This phenomenon seems to be related to the concentration of measure and other probabilistic perspectives.


2. A matrix $A$ is positive if and only if all of its eigenvalues are positive. We write $A\ge 0$.

A 2-by-2 positive matrix is in the form $$A=\left[\begin{array}{cc}t+x& y+iz\\ y-iz& t-x\end{array}\right]$$ where $t\ge 0$ and $x^2+y^2+z^2\le t^2$.

The matrix $$\left[\begin{array}{cc}{t}_0+x_0& y_0+i{z}_0\\ y_0-i{z}_0& t_0-x_0\end{array}\right]$$ can be identified with the ball
$$B(x_0,y_0,z_0;t_0):=\{(x,y,z)\in {\mathbb{R}}^3:(x-x_0{)}^2+(y-y_0{)}^2+(z-z_0{)}^2\le t_0^2\}.$$

Let $A_j=\left[\begin{array}{cc}{t}_j+x_j& y_j+i{z}_j\\ y_j-i{z}_j& t_j-x_j\end{array}\right]$ for $j=1,2$. We have the following equivalence:
$$\begin{array}{rl}& A_1\ge A_2\\ ⟺& (x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_1-z_2{)}^2\le (t_1-t_2{)}^2\\ & \text{ and }{t}_1\ge t_2\\ ⟺& B(x_2,y_2,z_2;t_2)\subset B(x_1,y_1,z_1;t_1).\end{array}$$
In other words, the ordering of the matrices corresponds to the inclusion of the balls!