# 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. Jun 19 '19 at 18:29
• @Clement $H_2$ is second homology Aug 10 '19 at 14:11