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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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