# 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.