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

The conjecture that I proposed in

http://mathoverflow.net/questions/78486/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 associatedto 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 2011 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 2011 at 19:38