projective plane

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

Partitioning the Projective Plane ★★

Author(s): Noel

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

\begin{definition} Say that a subset $S$ of the projective plane is \emph{octahedral} if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin. \end{definition}

\begin{definition} Say that a subset $S$ of the projective plane is \emph{weakly octahedral} if every set $S'\subseteq S$ such that $|S'|=3$ is octahedral. \end{definition}

\begin{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. \end{conjecture}

Keywords: Partitioning; projective plane

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