Generalization of the "double cap conjecture" to a vector space with complex field

Question

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"?

Answer 1

The conjecture is true for $n=2$, but I believe that solving this for $n>2$ is likely harder than solving the usual Double Cap Conjecture in $\mathbb{R}^n$.

Carefully calculating the volume of the set above, in $\mathbb{C}^n$ the double cap has density $\frac{1}{2^{n-1}}$ in the complex unit sphere. (This is calculated in the following paper of the OP https://arxiv.org/abs/1110.5944).

For $n=2$, this set has density $\frac{1}{2}$, and this is exact. Indeed, suppose that $U$ is a subset of the sphere. Given any two orthogonal vectors $z,w\in\mathbb{C}^2$, only one can be contained in $U$, and so, we find that the density of $U$ is at most $\frac{1}{2}$. Similarly, in dimension $n$, this argument yields an upper bound of $\frac{1}{n}$. This argument uses only real vectors, and is originally due to Witsenhausen.

For higher $n$, the arguments of Frankl-Wilson and Raigorodskii for the real case apply to the complex space directly, as did Witsenhausen's argument above, and these yield an asymptotic upper bound of the form $\frac{1}{1.15^n}$. However, this is substantially worse than in the real case, where asymptotically, the lower bound given by the double cap has size $\frac{1}{1.4142^n}$ instead of $\frac{1}{2^n}$.

I do not know of any way to modify Frankl-Wilson or Raigorodskii's argument to obtain better bounds for the complex case. One would naively think this should be possible, but satisfying an additional equation given by the complex parts causes problems when applying the polynomial method.

Despite the elegant lower bound of $\frac{1}{2^{n-1}}$, the gap between upper and lower bounds for the Complex Double Cap conjecture, both for $n=3$ and asymptotically, is substantially larger than for the real case.