Double-cap conjecture for Witsenhausen’s problem

Determine whether the maximum measure of a measurable subset of the unit sphere that contains no orthogonal pair equals twice the measure of a spherical cap of angular radius \(\pi/4\), equivalently whether the optimal value is \((1/\sqrt{2}+o(1))^n\) asymptotically.

Background

Witsenhausen’s problem asks for the largest fraction of the (n−1)(n-1)-dimensional unit sphere that can be covered by a measurable set containing no orthogonal pair. The union of two antipodal spherical caps of angular radius π/4\pi/4 gives a natural construction and hence a lower bound.

The dissertation notes that the proposed value has been confirmed only in dimension n=2n=2. Establishing the conjectured optimality of the double-cap construction would determine the extremal density in all dimensions and resolve the principal unresolved question associated with Witsenhausen’s problem.

References

It is conjectured that the optimal value is given by twice the measure of a spherical cap of angular radius~$\pi/4$Conjecture 2.8, which is~$(1/\sqrt{2} + o(1))n$. This has only been confirmed for~$n=2$.

— Optimization hierarchies for extremal geometry through complete positivity  (2609.34845 - Bekker, 28 Sep 2026) in Chapter 1, Introduction, discussion of Witsenhausen’s problem

The constraint ``$K(x, y) = 0$ if~${e, x, y}$ is not independent'' can therefore not be written equivalently in terms of such an expansion, hence it is unclear that the resulting problem would give an upper bound to~$\alpha_n$.

— Optimization hierarchies for extremal geometry through complete positivity  (2609.34845 - Bekker, 28 Sep 2026) in Chapter 7, Section 7.3, “The failure of the block moment hierarchy”