# Maximizing an integral w.r.t. a measure on the unit sphere

I would like to know if the answer to the following question is known.

Let $$d \ge 3$$. What is the value of $$\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \wedge \cdots \wedge x_d| \, d\mu(x_1) \cdots d\mu(x_d),$$ where the maximum is over all probability measures $$\mu$$ on the unit sphere $$S^{d−1}$$?

The notation $$x_1 \wedge \cdots \wedge x_d$$ stands for $$d$$-polyvectors (as usual in Geometric Measure Theory).

For $$d=2$$ the answer is in the paper Tilli, P., Isoperimetric inequalities for convex hulls and related questions. Trans. Amer. Math. Soc. 362 (2010), 4497-4509.

The techniques of that paper do not apply to the case $$d \ge 3$$ and I am curious if during the last ten years the problem has been approached by someone else.