  We introduce a conjecture that we call the {\it Two Hyperplane Conjecture},
saying that an isoperimetric surface that divides a convex body in half by
volume is trapped between parallel hyperplanes. The conjecture is motivated by
an approach we propose to the {\it Hots Spots Conjecture} of J. Rauch using
deformation and Lipschitz bounds for level sets of eigenfunctions. We will
relate this approach to quantitative connectivity properties of level sets of
solutions to elliptic variational problems, including isoperimetric
inequalities, Poincar\'e inequalities, Harnack inequalities, and NTA
(non-tangentially accessibility). This paper mostly asks questions rather than
answering them, while recasting known results in a new light. Its main theme is
that the level sets of least energy solutions to scalar variational problems
should be as simple as possible.
