Reformulated concurrent normals conjecture in T^*S^{n−1}

Show that for a convex body C ⊂ R^n with smooth boundary, there exists a point q* ∈ C such that the Lagrangian submanifolds L_{q*} = graph(d⟨q*,−⟩) and L_{∂C} = graph(dh_C) in T^*S^{n−1} intersect in exactly 2n points; equivalently, establish the existence of q* with #(L_{q*} ∩ L_{∂C}) = 2n.

Background

The space of oriented lines in R^n is symplectomorphic to T^*S^{n−1}, where graphical Lagrangians represent families of oriented normals: L_q corresponds to lines normal to a point q, and L_{∂C} corresponds to normals to ∂C via the support function h_C. The classical concurrent normals conjecture can be recast as a Lagrangian intersection problem in T^*S^{n−1}.

This symplectic reformulation fits the paper’s broader theme that surplusection (excess intersections over the minimal possible) is intrinsic in many settings. Proving the conjecture in this form would tie convex geometric caustics directly to Lagrangian intersection theory.