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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 ...


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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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Let $R\subseteq P$ be the region $\{(x,y)\in P:x>0,y>0, x+y>1\}$ and let $g:P\to R$ be continuous and bijective. Let $h(p_2, p_3, p_4; (x, y)) = p_3+(p_2-p_3)y + (p_4-p_3)x$. Note that $h: P^4 \to P$ is continuous, and that $h(p_2, p_3, p_4; q_i)=p_i$, where $q_2=(0,1)$, $q_3=(0,0)$, and $q_4=(1,0)$. Let $f_1(p_1,p_2,p_3,p_4) = h(p_2, p_3, p_4; g(...


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