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

\displaystyle 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 \rightarrow 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

\displaystyle 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

\displaystyle \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

\displaystyle 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:

\displaystyle \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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