# Concentration properties of inner-products in high-dimension

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 $$q = , x \in A$$. 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 $$d << K$$)?

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.