  We prove ``half-space" intersection properties in three settings: the
hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian
space. For instance, any two embedded minimal hypersurfaces in the sphere must
intersect in every closed hemisphere. Two approaches are developed: one using
classifications of stable minimal hypersurfaces, and the second using conformal
change and comparison geometry for $\alpha$-Bakry-\'{E}mery-Ricci curvature.
  Our methods yield the analogous intersection properties for free boundary
minimal hypersurfaces in space form balls, even when the interior or boundary
curvature may be negative.
  Finally, Colding and Minicozzi recently showed that any two embedded
shrinkers of dimension $n$ must intersect in a large enough Euclidean ball of
radius $R(n)$. We show that $R(n) \leq 2 \sqrt{n}$.
