6 problems

Subset-normal bound conjecture. The inequalities above hold.

Facet-normal characterization conjecture. The normals of $K$ are the same as either those of a simplex bounded by an extra facet, or those of a pyramid that is not a simplex.

Let $(\operatorname{ICP})$ denote the integral Carathéodory property for normal polytopes, and consider counterexamples to this property in varying dimensions. Trivial extensions a…

Asymptotic Carathéodory-rank conjecture. The ratio $\operatorname{CR}(d)/d$ is monotonically converging to $2$ as $d\to\infty$. The source records a counterexample to the integral…

Let $P\subset\mathbb{R}^d$ be a normal lattice polytope. For every $c\in\mathbb{N}$ and every $x\in(cP)\cap\mathbb{Z}^d$, consider representations of $x$ as a weighted sum of latti…

Integral Carathéodory conjecture. Every rational finite pointed cone has the integral Carathéodory property. The polytope formulation is the corresponding special case under homoge…