![](/files/happy5.png)
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.
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ \pi/4 $](/files/tex/01608ea3b80f85b77096d16610a43e184782386c.png)
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Partitioning the Projective Plane ★★
Author(s): Noel
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset
of the projective plane is octahedral if all lines in
pass through the closure of two opposite faces of a regular octahedron centered at the origin.
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
Definition Say that a subset
of the projective plane is weakly octahedral if every set
such that
is octahedral.
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S'\subseteq S $](/files/tex/e8109a0f5c1dcb11fe3245b53b8bd2bc9d6418d1.png)
![$ |S'|=3 $](/files/tex/e1b8bc4df405ab37c6b4aa2f638a5c2df882f7a9.png)
Conjecture Suppose that the projective plane can be partitioned into four sets, say
and
such that each set
is weakly octahedral. Then each
is octahedral.
![$ S_1,S_2,S_3 $](/files/tex/6f2284a502ee75f719fa3d5c2430c467e11df0c4.png)
![$ S_4 $](/files/tex/a0fc8ce0b0dfbf88309c7c045fff90a5cadd5117.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
Keywords: Partitioning; projective plane
![Syndicate content Syndicate content](/misc/feed.png)