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A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh, Ivan Hetman (2012)

Studia Mathematica

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 C o n v ( X ) of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex...

A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria

Pedro A. Góngora, David A. Rosenblueth (2015)

International Journal of Applied Mathematics and Computer Science

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...

An inequality concerning edges of minor weight in convex 3-polytopes

Igor Fabrici, Stanislav Jendrol' (1996)

Discussiones Mathematicae Graph Theory

Let e i j 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 , 6 120 ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

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