Friday, April 11, 2014

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 $$B(x;R)=\{ (x_1, \cdots, x_N) : x_1^2 + \cdots + x_N^2 \leq R^2 \} . $$ The fraction of the volume of $B(x; \alpha R)$ to the volume of $B(x; R)$ is $\alpha^N$, and $\lim_{N \to 0} \alpha^N = 0$. It means that "a full sphere of high dimension has all its volume within a "skin" of size $\varepsilon$ 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 $A \geq 0$.

A 2-by-2 positive matrix is in the form $$A = \begin{bmatrix} t+x & y + iz \\ y-iz & t-x \end{bmatrix}$$ where $t \geq 0$ and $x^2 + y^2 + z^2 \leq t^2$.

The matrix $$ \begin{bmatrix} t_0+x_0 & y_0 + i z_0 \\ y_0-i z_0 & t_0-x_0 \end{bmatrix} $$ can be identified with the ball
$$ B(x_0,y_0,z_0;t_0) := \{ (x,y,z) \in \mathbb{R}^3 : (x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 \leq t_0^2 \} . $$

Let $A_j = \begin{bmatrix} t_j +x_j & y_j + i z_j \\ y_j - i z_j & t_j - x_j \end{bmatrix}$ for $j=1,2$. We have the following equivalence:
$$\begin{array}{rl}
& A_1 \geq A_2 \\
\Longleftrightarrow & (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2 \leq (t_1 - t_2)^2 \\
& \text{ and } t_1 \geq t_2 \\
\Longleftrightarrow & B(x_2,y_2,z_2;t_2) \subset B(x_1,y_1,z_1;t_1) .
\end{array} $$
In other words, the ordering of the matrices corresponds to the inclusion of the balls!


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