### 1-bend 3-D orthogonal box-drawings: Two open problems solved.

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A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space $Con{v}_{}\left(X\right)$ of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex...

Consider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending...

Let ${e}_{ij}$ be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is $20{e}_{3,3}+25{e}_{3,4}+16{e}_{3,5}+10{e}_{3,6}+6[2/3]{e}_{3,7}+5{e}_{3,8}+2[1/2]{e}_{3,9}+2{e}_{3,10}+16[2/3]{e}_{4,4}+11{e}_{4,5}+5{e}_{4,6}+1[2/3]{e}_{4,7}+5[1/3]{e}_{5,5}+2{e}_{5,6}\ge 120$; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.