Concurrent normals conjectureGeneralization of the non-existence of a monostatic planar bodyA question of...
Concurrent normals conjecture
Generalization of the non-existence of a monostatic planar bodyA question of compactness in the geometry of numbersA question on the Mahler conjectureAlgorithmic Version of John's Decomposition of Convex BodyUnit ball of smallest volume in a Hilbert geometryAn affine invariant of convex bodiesStatus of Barany's conjecture?Geodesics on convex hypersufacesCake cutting conjectureMinkowski sum of polytopes from their facet normals and volumes
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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 (2012), no. 1, 55–71
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?
convex-geometry
asked 8 hours ago
ClementClement
362
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add a comment |
$begingroup$
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 (2012), no. 1, 55–71
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?
convex-geometry
asked 8 hours ago
ClementClement
362
$endgroup$
add a comment |
$begingroup$
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 (2012), no. 1, 55–71
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?
convex-geometry
asked 8 hours ago
ClementClement
362
$endgroup$
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 (2012), no. 1, 55–71
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?
convex-geometry
convex-geometry
asked 8 hours ago
ClementClement
362
asked 8 hours ago
ClementClement
362
asked 8 hours ago
ClementClement
362
asked 8 hours ago
ClementClement
362
asked 8 hours ago
ClementClement
362
362
add a comment |
add a comment |
1 Answer
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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 $Ksubseteqmathbb R^n$ with closure $overline K$, and we consider the "distance to $p$" function $d_p:partial Ktomathbb R$ for $pinoverline K$. Let $Vsubseteqoverline K$ be the set of $pinoverline K$ 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 $pinoverline K$, 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.
answered 8 hours ago
John PardonJohn Pardon
9,650333108
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $Ksubseteqmathbb R^n$ with closure $overline K$, and we consider the "distance to $p$" function $d_p:partial Ktomathbb R$ for $pinoverline K$. Let $Vsubseteqoverline K$ be the set of $pinoverline K$ 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 $pinoverline K$, 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.
answered 8 hours ago
John PardonJohn Pardon
9,650333108
$endgroup$
add a comment |
$begingroup$
Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $Ksubseteqmathbb R^n$ with closure $overline K$, and we consider the "distance to $p$" function $d_p:partial Ktomathbb R$ for $pinoverline K$. Let $Vsubseteqoverline K$ be the set of $pinoverline K$ 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 $pinoverline K$, 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.
answered 8 hours ago
John PardonJohn Pardon
9,650333108
$endgroup$
add a comment |
$begingroup$
Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $Ksubseteqmathbb R^n$ with closure $overline K$, and we consider the "distance to $p$" function $d_p:partial Ktomathbb R$ for $pinoverline K$. Let $Vsubseteqoverline K$ be the set of $pinoverline K$ 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 $pinoverline K$, 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.
answered 8 hours ago
John PardonJohn Pardon
9,650333108
$endgroup$
Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $Ksubseteqmathbb R^n$ with closure $overline K$, and we consider the "distance to $p$" function $d_p:partial Ktomathbb R$ for $pinoverline K$. Let $Vsubseteqoverline K$ be the set of $pinoverline K$ 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 $pinoverline K$, 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.
answered 8 hours ago
John PardonJohn Pardon
9,650333108
answered 8 hours ago
John PardonJohn Pardon
9,650333108
answered 8 hours ago
John PardonJohn Pardon
9,650333108
answered 8 hours ago
John PardonJohn Pardon
9,650333108
9,650333108
add a comment |
add a comment |
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