15 problems

Conway–Guy conjecture. There is no monostable tetrahedron, but there is a monostable convex polyhedron in $\mathbb R^3$.

Higher-dimensional parallelogram decomposition conjecture. Every surface in

Let $P$ be a convex polyhedron. A simple closed geodesic is a simple closed geodesic on the surface of $P$, and a simple closed quasigeodesic through exactly one vertex is a simple…

Continuous blooming conjecture. Every convex polyhedral boundary has a continuous blooming.

Equivistal subdivision conjecture. The equivalence relation induced by equivistality constitutes a convex polyhedral subdivision of $S$. Moreover, the number of open regions in thi…

Polynomial combinatorial-type conjecture. The cardinality of the set of combinatorial types of shortest paths in $S$ is polynomial in the number of facets of $S$ when the dimension…

Polynomial source-image conjecture. There is a fixed polynomial $f_d$, independent of both $S$ and $v$, such that

Parallelogram decomposition conjecture. Every centrally symmetric convex polyhedral surface can be decomposed into finitely many parallelograms.

Let $Z_1$, $Z_2$, and $Z_4$ denote the conditions referenced in the theorem above, and let the theorem assert the existence result under the stronger condition $Z_4$. Weaker-condit…

Let the geodesic complexity of a convex polyhedron mean the minimum number of sets needed to cover its geodesic motion-planning problem, so that a complexity of at most four means…

Let the geodesic complexity of a convex polyhedron mean the minimum number of sets needed to cover its geodesic motion-planning problem, so that a complexity of at least four means…

Endpoint-emptiness conjecture. The polyhedron $P(csigma_P(\lambda))$ must be empty.

Grünbaum's conjecture. Every convex polyhedron is unfoldable.

Let $P$ be a convex polyhedron. A steepest edge tree is a spanning tree of the edge graph of $P$ obtained by the steepest-edge algorithm, and a tree generates a simple unfolding wh…

Small-genus convex-face conjecture. No regular maps other than the eight currently known examples admit polyhedral embeddings with convex faces.