Imre Bárány

Notable results

• He gave a surprisingly simple alternative proof of László Lovász's theorem on Kneser graphs.
• He gave a new proof to the Borsuk–Ulam theorem.
• Barany gave a colored version of Carathéodory's theorem.
• He solved an old problem of James Joseph Sylvester on the probability of random point sets in convex position.
• With Van H. Vu proved a central limit theorem on random points in convex bodies.
• With Zoltán Füredi he gave an algorithm for mental poker.
• With Füredi he proved that no deterministic polynomial time algorithm determines the volume of convex bodies in dimension d within a multiplicative error d^d.
• With Füredi and János Pach he proved the following six circle conjecture of László Fejes Tóth: if in a planar circle packing each circle is tangent to at least 6 other circles, then either it is the hexagonal system of circles with identical radii, or there are circles with arbitrarily small radius.

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