Concentration properties of inner-products in high-dimension

Question

Let $S^K$ be the unit sphere embedded in $R^{K+1}$.

$v \in S^K$ is randomly chosen from a uniform distribution over $S^K$.

$A \subseteq S^K$ is a $d$-dimensional sub-manifold ($d \leq K$). Think of A as a patch of $S^K$. $A$ may or may not have a boundary.

I want to compute the distrbution of the inner-products of $v$ and the points in $A$, i.e., the distribution $P(q)$ such that $q = <x,v>, x \in A$. What can we say about the concentration properties of $P(q)$? How does it depend on $vol(A)$ ? By $vol(A)$, I mean the $d$-dimensional volume measure of $A$.

When $A = S^K$, it is well studied; the distribution concentrates very sharply around $q=0$, because most of the points are orthogonal w.r.to $v$. but how does one go about reasoning for smaller sub-manifolds (especially when $d << K$)?

Answer 1

Since you have the same concentration properties when $d=0,$ and $A$ is a point (by rotation invariance), the answer is Yes, there is always concentration.