6 problems

Finite-generation conjecture. For every positive integer $n$, this closed cone is finitely generated.

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegativ…

Cone equality conjecture. The two cones are equal:

Let $P$ be a centrally symmetric $d$-dimensional polytope. A flag of $P$ is a chain of faces of $P$, one in each dimension. Kalai's full flag conjecture. $P$ has at least…

Let $P_{01,d}$ be the set of all $d$-dimensional $01$-polytopes, together with their polars. A set $P$ of $d$-polytopes is broad when every word-set $bb_s$ that is nonnegative on e…

Let $X$ be a convex polytope, and let the extended $g$-vector be the vector whose components are the linear functions on flag vectors specified by the axioms in the previous sectio…