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$\newcommand{\1}{\mathbf 1}\newcommand{\ep}{\varepsilon}\newcommand{\tr}{\operatorname{tr}}$The min-max value is $\sqrt n$. Indeed, take any real $n\times n$ matrix $H$ with $|\det H|=1$. By the singular value decomposition, \begin{equation} H=U^TDV, \end{equation} where $U$ and $V$ are some orthogonal matrices and $D$ is the diagonal matrix with ...


The conditions you pose on $P_0$ imply that it is a reflexive polytope. (That is, a lattice polytope with the origin in its interior and such that its polar dual is also a lattice polytope). There are finitely many reflexive polytopes in each dimension (modulo $GL(\mathbb Z,n)$), which implies that the answer to your question is negative. For example, in ...


One possible definition is that a concrete polytope is an abstract polytope where each vertex has been assigned to a point in space, and all elements of rank n are contained in nD subspaces.

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