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3 votes
1 answer
156 views

Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
4 votes
0 answers
93 views

On symmetry and measure concentration rate for convex bodies

The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$, states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $, we ...
4 votes
1 answer
290 views

On the 1/2 assumption on concentration of measure for continuous cube

The concentration of measure on $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$, states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $, we have: $$...
2 votes
0 answers
60 views

Mean width of intersection of two elipsoid

My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width \begin{align*} \omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
10 votes
2 answers
847 views

Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{...