# Covering a sphere with ellipsoid-products in high dimension

For $$\Sigma\geq 0$$ a $$k\times k$$ matrix and large $$n$$, fix $$E:= \{(x_i)_{i=1}^n: \sum_i x_i^\dagger \Sigma x_i \leq n\}$$. Fix $$(z_m)_m$$ as $$M$$ points iid uniform on $$\mathbb{S}^{nk-1}\subset \mathbb{R}^{nk}.$$ How large must $$M$$ be for $$\cup_m (z_m+ E)$$ to likely cover $$\mathbb{S}^{nk-1}$$?

Similarly, how small must $$M$$ be so that overlaps, $$\cup_{(i,j)}(z_i+E)\cap (z_j+E)$$, normalized to $$\mathbb{S}^{nk-1}$$'s area, likely have small volume?