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Combinatorics

The Double Cap Conjecture ★★

Author(s): Kalai

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.

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

Posted by Jon Noel
updated September 15th, 2015
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Geometry

Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in $ \mathbb{R}^3 $.

Definition   Say that a subset $ S $ of the projective plane is octahedral if all lines in $ S $ pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset $ S $ of the projective plane is weakly octahedral if every set $ S'\subseteq S $ such that $ |S'|=3 $ is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say $ S_1,S_2,S_3 $ and $ S_4 $ such that each set $ S_i $ is weakly octahedral. Then each $ S_i $ is octahedral.

Keywords: Partitioning; projective plane

Posted by Jon Noel
updated August 27th, 2013
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