Concurrent normals conjecture (Problem A3)

Prove that for any smooth convex body C ⊂ R^n, there exists a point q ∈ C such that q lies on at least 2n inward normals to the boundary ∂C.

Background

The authors discuss the geometry of normals to convex bodies and the existence of caustics with multiplicities at least as large as in the ellipsoid case. The conjecture asserts that every convex body in Rn has a point through which at least 2n boundary normals pass.

The conjecture has been proven in dimensions 2, 3, and 4 and for centrally symmetric bodies, but remains open in higher dimensions and in full generality. The paper connects this problem to Lagrangian surplusection phenomena, motivating further analysis via symplectic methods.

References

Conjecture [Problem A3] If C is a convex body in Rn (with smooth boundary for the purposes of our discussion), then there exists a point q\in C such that q lies on at least 2n inward normals to \partial C.

— Lagrangian Surplusection Phenomena (2408.14883 - Rizell et al., 27 Aug 2024) in Section 3.1 (The original conjecture)