Exact value of the discretized Kakeya needle constant at n=5

Determine the exact value of \(C_T(5)\), the minimum union area of five horizontally translatable thin triangles representing five equally spaced directions in the discretized Kakeya needle problem.

Background

The paper studies the finite Kakeya needle quantity C_T(n), defined as the infimum of the area of a union of n direction-dependent triangles over horizontal offsets. It proves exact values for n=3 and n=4 and shows that every global minimizer for n=5 must be asymmetric.

An asymmetric construction gives area 14/61, improving the best reflection-symmetric value 7/30, but the global optimum is not determined.

References

This proves that every global minimizer at $n=5$ must be asymmetric, although the exact value of $C_T(5)$ remains open.

— Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment  (2608.23691 - Chung et al., 24 Aug 2026) in Section 3, subsection “Discretized Kakeya needle,” paragraph S2