Concurrent normals conjecture for convex bodies in R^n

Establish that for every integer n ≥ 2 and every compact convex body P contained in R^n, there exists a point y in the interior of P that lies on at least 2n distinct inward normals to the boundary ∂P, each normal having a different base point on ∂P.

Background

The paper opens by recalling a long-standing conjecture in the smooth setting relating normals to convex bodies: for any convex body in R^n, there should exist an interior point lying on at least 2n boundary normals. This classical conjecture is known to hold in low dimensions (n = 2, 3, 4), but remains unresolved in general.

The authors’ work focuses on the polyhedral (piecewise-linear) counterpart of this problem, connecting the count of normals to critical points of the squared distance function within a Morse-theoretic framework. Their results highlight differences between smooth bodies and polytopes, but they cite the smooth conjecture as a motivating open problem.