1. “Take a sphere of radius in dimensions, large; then most points inside the sphere are in fact very close to the surface.” David Ruelle
Let and . Let
The fraction of the volume of to the volume of is , and It means that “a full sphere of high dimension has all its volume within a “skin” of size near the surface” (Collet and Eckmann). This phenomenon seems to be related to the concentration of measure and other probabilistic perspectives.
2. A matrix is positive if and only if all of its eigenvalues are positive. We write .
A 2-by-2 positive matrix is in the form
where and .
can be identified with the ball
Let for . We have the following equivalence:
In other words, the ordering of the matrices corresponds to the inclusion of the balls!