The Real Polarization Problem

Let $d\geq 2$, let $x_1,\ldots,x_d$ be unit vectors in $\mathbb{R}^d$, and let $v$ be a unit vector in $\mathbb{R}^d$. The $d$th linear polarization constant $c_d(\mathcal{H})$ is defined for a real Hilbert space $\mathcal{H}$ by

$$c_d(\mathcal{H})=\inf\left\{M>0:\ \forall u_1,\ldots,u_d\in S(\mathcal{H}),\ \exists v\in S(\mathcal{H}):\left|\langle u_1,v\rangle_{\mathcal{H}}\cdots\langle u_d,v\rangle_{\mathcal{H}}\right|\geq M^{-1}\right\},$$

where $S(\mathcal{H})=\{u\in\mathcal{H}:\|u\|_{\mathcal{H}}=1\}$. Real Polarization Problem. For any $d\geq 2$ and any collection $x_1,\ldots,x_d$ of unit vectors in $\mathbb{R}^d$, there exists a unit vector $v\in\mathbb{R}^d$ such that

$$\left|\langle v,x_1\rangle\cdots\langle v,x_d\rangle\right|\geq d^{-d/2}.$$

As a consequence, for $d\geq 2$ and every real Hilbert space $\mathcal{H}$ of dimension at least $d$, one has $c_d(\mathcal{H})=d^{d/2}$. The complex analogue is known, while this real version is proved only for $d\leq 5$ and remains open in the remaining cases.