I see. Thank you very much for the answers and the exposition. The game-theoretic proof of Menger’s theorem is fascinating.

]]>Oops, there was a typo in this paragraph; it is assumed that Alice and Bob are both playing optimal strategies (not just Alice). Since Alice is guaranteed a payoff of no matter what strategy Bob plays, every possible path has to be hit with probability at least ; but the optimal strategy that Bob actually is playing gives a payoff of exactly , so all the paths that Bob uses also has to be hit with probability exactly (otherwise if one of these paths was hit with a larger probability, the payoff would be strictly larger than ).

]]>Would anybody be able to give a hint as to why this is true?

]]>I’ve actually asked other people (such as Keith Ball) about this, since I do not have much native experience with high-dimensional convex bodies. But the viewpoint of Vitali Milman that was pointed out by Gil Kalai in the same math overflow post ( https://mathoverflow.net/a/26010/766 ) is, I think, a good one: convex bodies in high dimensions are largely ellipsoidal but also can contain a number of “spikes” that are small in volume but which can protrude well beyond the main ellipsoid of the convex body. Foundational results in high dimensional convex geometry such as Dvoretsky’s theorem, the Bourgain-Tzafriri theorem, or the Johnson-Lindenstrauss lemma are broadly in line with this intuitive picture. (For low dimensions one has instead John’s ellipsoid theorem, which basically says that one just has the ellipsoid and no spikes.)

]]>You need one of those darn “latex” commands here.

*[Corrected, thanks – T.]*

MI PREGUNTA ES

¿ SE RESTRINGE ESTE TIPO DE ANÁLISIS O EXTENSIÓN EN LA TEORÍA DE REPRESENTACIÓN DE CONJUNTOS PARCIALMENTE ORDENADOS EN EL CASO DE POSET CRÍTICOS ( KLEINER O NAZAROVA) O SE GENERA UN PUENTE NETAMENTE ALGEBRAICO? ]]>

Ah, I see, thank you.

]]>When we apply the induction hypothesis to the system , , the coefficients generated by conclusion 2 of Farkas’s lemma applied in dimension d-1 do not depend on the underlying variables .

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