# Generalization of the “double cap conjecture” to a vector space with complex field

The conjecture that I proposed in

Maximal set on hypersphere that does not contain pairs of orthogonal vectors

is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun. See for example

http://gilkalai.wordpress.com/2009/05/22/how-large-can-a-spherical-set-without-two-orthogonal-vectors-be/

Here I propose a generalization. Suppose that the field of the unit vectors is not real, but complex. We can define a measure, which is induced by the measure of a real vector space. For example, the two-dimensional complex vectors $(a,b)$ are associated to the 4-dimensional real vectors $(\Re(a),\Re(b),\Im(a),\Im(b))$. Thus, the measure on the latter space induces a measure on the former. What is the set of unit vectors with maximal area that does not contain pairs of orthogonal vectors. My conjecture: up to unitary rotations, the maximal set is the set of vectors $a$ such that $|a\cdot S|^2>1/2$, where $S$ is some unit vector. This is a generalization of the "double cap conjecture".

The question is: is there a easy proof of this conjecture from the "double cap conjecture"?

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Nice conjecture! – Gil Kalai Oct 21 '11 at 18:46
Notice that the maximal set is a set of rays, that is, if $x\in M$ and $M$ is the maximal set, then $a x\in M$, where $a$ is any complex number of modulus 1. – Alberto Montina Oct 22 '11 at 19:38