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Let $x$ be a vertex of $\mathcal{P}$. By Cramer's rule, there is an $n \times n$ matrix $C$ such that each coordinate of $x$ is an integer multiple of $\frac{1}{|\det(C)|}$, and the absolute value of …

By the Upper Bound Theorem, the maximum number of $(d-1)$-dimensional faces of an $n$-vertex polytope is achieved by the cyclic polytopes. This number can be explicitly written via the Dehn-Sommervil …

Your question is essentially about extension complexity. In general, the extension complexity of a polytope $P$ is the minimum number of facets over all polytopes $Q$ which project to $P$. You are i …

It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $. In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily colum …

The answer to the third question is no. This is a rather counter-intuitive discovery of Micha Perles from the sixties. See this paper of Ziegler, for a simpler construction and other pertinent infor …

Here is an explicit example where the number of facets can increase dramatically when removing a vertex from the convex hull. Let $G$ be a graph and define the subgraph polytope $P(G)$ of $G$ to be t …

This question is a bit vague, but you may be looking for Motzkin's decomposition theorem. This theorem says that any polyhedral convex set can be expressed as the Minkowski sum of a polytope and a po …

For the lower bound, I am guessing that every set of N points in the plane contains at least N-1 legal subsets, with equality if and only if the points are in a line.