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Kalai, Gil
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 $](/files/tex/01608ea3b80f85b77096d16610a43e184782386c.png)
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Cube-Simplex conjecture ★★★
Author(s): Kalai
Conjecture For every positive integer
, there exists an integer
so that every polytope of dimension
has a
-dimensional face which is either a simplex or is combinatorially isomorphic to a
-dimensional cube.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ \ge d $](/files/tex/616a25c014115485d010b8d180f072cc22ed1c53.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
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