# projective plane

## 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

## Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by *projective plane* we mean the set of all lines through the origin in .

**Definition**Say that a subset of the projective plane is

*octahedral*if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.

**Definition**Say that a subset of the projective plane is

*weakly octahedral*if every set such that is octahedral.

**Conjecture**Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.

Keywords: Partitioning; projective plane