# Concurrent normals conjecture

It is conjectured that if $$K$$ is a convex body in $$n$$-dimensional Euclidean space, then there exists a point in the interior of $$K$$ which is the point of concurrency of normals from $$2n$$ points on the boundary of $$K$$. This has been proved for $$n=2$$ and $$3$$ by E. Heil. For $$n=4$$, a proof appeared (under a smoothness assumption on the boundary) in

Pardon, John, Concurrent normals to convex bodies and spaces of Morse functions, Math. Ann. 352, No. 1, 55-71 (2012). ZBL1242.52006.

but a reviewer’s remark in zbMATH says that in this paper, the proof of the first theorem "does not seem to be quite correct". So my question is:

What is the current status of this conjecture for $$n=4$$?

Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $$K\subseteq\mathbb R^n$$ with closure $$\overline K$$, and we consider the "distance to $$p$$" function $$d_p:\partial K\to\mathbb R$$ for $$p\in\overline K$$. Let $$V\subseteq\overline K$$ be the set of $$p\in\overline K$$ for which $$d_p$$ has exactly one local minimum. I claimed in my paper that "it is clear that $$V$$ is closed". As the reviewer correctly points out, this is false (counterexample: $$K$$ the unit ball). But here is a corrected version: "if $$d_p$$ has finitely many local minima for every $$p\in\overline K$$, then $$V$$ is closed". Indeed, if $$d_p$$ has finitely many local minima, then if we perturb $$p$$ the number of local minima can only increase. This corrected version of the statement is sufficient for the proof given in the paper, since if $$d_p$$ has infinitely many local minima for some $$p$$, this exactly means that there are infinitely many normals to $$\partial K$$ which are concurrent at $$p$$. So the reviewer's remark doesn't invalidate the argument.
• It follows immediately from the "Morse inequalities", but I prefer to think about it more concretely as follows. I claim that any Morse function $f:S^3\to\mathbb R$ with $k$ local minima must have at least $k-1$ index one critical points (this implies what we want since $3+2+2+1=8$). This is obvious if one thinks about how the topology of the sublevel sets changes when passing through a Morse critical point: every time we pass a local minimum the number of components increases, and the only way for the number of components to decrease is if we pass through an index one critical point. – John Pardon Jun 19 at 18:29