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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.

See also: https://mathoverflow.net/a/322726/121665 for some some other comments on Tilli's paper.

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