Random Sampling and the Vector in Subspace Problem

Let A be a set on the n dimensional sphere, and let H be a random subspace of dimension n/2.
What can we say about the measure of the intersection? How close is it to the measure of the original set A?

Klartag and Regev gave an optimal answer for this question for large enough sets.
We will discuss how considering both the intersection with H and with its orthogonal complement simultaneously may give a better result for smaller sets.
We prove a partial result under some additional geometric condition.

One motivation to these results is a communication complexity problem called Vector in Subspace Problem (VSP).