# Kalai, Gil

## The Double Cap Conjecture ★★

Author(s): Kalai

\begin{conjecture} The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\pi/4$ around the north and south poles. \end{conjecture}

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

## Cube-Simplex conjecture ★★★

Author(s): Kalai

\begin{conjecture} For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is either a simplex or is combinatorially isomorphic to a $k$-dimensional cube. \end{conjecture}