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.

I have to admit, this is not a well written paper, and I would be surprised if it did not contain a number of other equally badly presented arguments. I can say that I reread it a couple of years ago and was more or less convinced by the proof. However, I have not discussed it in detail with anyone.

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    $\begingroup$ Many thanks for the explanations. In the paper, you wrote that any Morse function in the 3-sphere with at least 3 local maxima and 2 local minima has at least 8 critical points. Is it a trivial remark? Or is it a consequence of the below theorem by Camacho and Scardua ? A Morse foliation F (of a compact connected manifold M) such that the number c of centers and the number s of saddles in Sing(F) satisfy c>s, then c=s+2 and M is homeomorphic to an dim(M)-sphere or c=s+1 and M is a Eells-Kuiper manifold. $\endgroup$ – Clement Jun 19 '19 at 12:34
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    $\begingroup$ 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. $\endgroup$ – John Pardon Jun 19 '19 at 18:29
  • $\begingroup$ Dear John, I am trying to understand your proof but I have difficulties.(may be from my inexperience of the subject). In your proof of Lemma 4.2, you wrote "...which has nonzero H2". But, it seems to me that H2 has not been defined before in the text (the one dated "17 December 2008")... Can you help me ? $\endgroup$ – Clement Aug 10 '19 at 12:50
  • $\begingroup$ @Clement $H_2$ is second homology $\endgroup$ – John Pardon Aug 10 '19 at 14:11
  • $\begingroup$ Dear John. In the proof of Lemma 3.3, you wrote "Then E is a bundle over B with fiber D²". Is it for the special case n=3 or is it true for any n ? $\endgroup$ – Clement Dec 28 '19 at 18:57

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