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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 $n=4$?

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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 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 at 18:29

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