Kalai, Gil
The Double Cap Conjecture ★★
Author(s): Kalai
Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Cube-Simplex conjecture ★★★
Author(s): Kalai
Conjecture For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.