Grünbaum’s uneven orthogonal partition conjecture for convex bodies
Determine whether every planar convex body of area one admits, for every target t with 0≤t≤1/4, two perpendicular lines whose four cyclically ordered regions have areas t, t, 1/2−t, and 1/2−t.
References
Theorem~\ref{thm:counterintro} settles B ar any's general-measure question, but Gr"unbaum's convex-body problem remains open. The corollaries in Section \ref{sec:convex} deal with triangles and trapezoids.
The next natural test case is that of general convex quadrilaterals, which remains open. We believe this family could yield new insights on the problem.
\begin{problem}\label{prob:quadri} Does every convex quadrilateral satisfy Gr\"unbaum's conjecture for every target? \end{problem}
Our counterexamples are constructed separately for each $t\in(0,1/4)$, so the possible behavior of the target set for a single measure remains open.
\begin{problem}\label{q:targets} For a fixed line-null probability measure $\mu$, let
T(\mu)= \left{ t\in[0,1/4]: \text{$\mu$ admits the prescribed orthogonal partition at target $t$} \right}.
What sets can occur as $T(\mu)$? What additional structure does $T(\mu)$ have for regular measures, or for uniform measures on convex bodies? \end{problem}