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Dmitrii Korshunov
Dmitrii Korshunov
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Characterization of convexity by connectivityconnectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.

Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is connected?

If this is the case and is (surely) known I would appreciate the reference.

I have a sort of heuristic argument for $n=3$:

Suppose $p\in S$ is a point with strictly negative curvature, so the surface near $p$ qualitatively looks like athe graph of $X^2-Y^2$ around $0$ (up to the second order). Consider the tangent hyperplane $H_p$ at $p$. If necessary, perturb $p$ slightly to make sure that $H_p\setminus\{p\}$ intersects $S$ transversally (i.e. $H_p\cap S$ is singular only at $ p$). Now if we slightly parallel tranport $H_p$ up and down, the corresponding intersections $H^+_p\cap S$ and $H^-_p\cap S$ will look near $p$ like this:

enter image description here

with the rest of the picture being topologically unaltered. Case by case analysis suggests that $H^-_p\cap S$ and $H^+_p\cap S$ cannot both be conneted: indeed, the strings can be connected 1-4;2-3 or 1-2;4-3 (1-3;2-4 would cause intersections other than $p$).

Characterization of convexity by connectivity of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.

Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is connected?

If this is the case and is (surely) known I would appreciate the reference.

I have a sort of heuristic argument for $n=3$:

Suppose $p\in S$ is a point with strictly negative curvature, so the surface near $p$ looks like a graph of $X^2-Y^2$ around $0$ (up to the second order). Consider the tangent hyperplane $H_p$ at $p$. If necessary, perturb $p$ slightly to make sure that $H_p\setminus\{p\}$ intersects $S$ transversally (i.e. $H_p\cap S$ is singular only at $ p$). Now if we slightly parallel tranport $H_p$ up and down, the corresponding intersections $H^+_p\cap S$ and $H^-_p\cap S$ will look near $p$ like this:

enter image description here

with the rest of the picture being topologically unaltered. Case by case analysis suggests that $H^-_p\cap S$ and $H^+_p\cap S$ cannot both be conneted: indeed, the strings can be connected 1-4;2-3 or 1-2;4-3 (1-3;2-4 would cause intersections other than $p$).

Characterization of convexity by connectedness of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.

Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is connected?

If this is the case and is (surely) known I would appreciate the reference.

I have a sort of heuristic argument for $n=3$:

Suppose $p\in S$ is a point with strictly negative curvature, so the surface near $p$ qualitatively looks like the graph of $X^2-Y^2$ around $0$ (up to the second order). Consider the tangent hyperplane $H_p$ at $p$. If necessary, perturb $p$ slightly to make sure that $H_p\setminus\{p\}$ intersects $S$ transversally (i.e. $H_p\cap S$ is singular only at $ p$). Now if we slightly parallel tranport $H_p$ up and down, the corresponding intersections $H^+_p\cap S$ and $H^-_p\cap S$ will look near $p$ like this:

enter image description here

with the rest of the picture being topologically unaltered. Case by case analysis suggests that $H^-_p\cap S$ and $H^+_p\cap S$ cannot both be conneted: indeed, the strings can be connected 1-4;2-3 or 1-2;4-3 (1-3;2-4 would cause intersections other than $p$).

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Dmitrii Korshunov
Dmitrii Korshunov
  • 2.9k
  • 20
  • 27

Characterization of convexity by connectivity of hyperplane sections

Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.

Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is connected?

If this is the case and is (surely) known I would appreciate the reference.

I have a sort of heuristic argument for $n=3$:

Suppose $p\in S$ is a point with strictly negative curvature, so the surface near $p$ looks like a graph of $X^2-Y^2$ around $0$ (up to the second order). Consider the tangent hyperplane $H_p$ at $p$. If necessary, perturb $p$ slightly to make sure that $H_p\setminus\{p\}$ intersects $S$ transversally (i.e. $H_p\cap S$ is singular only at $ p$). Now if we slightly parallel tranport $H_p$ up and down, the corresponding intersections $H^+_p\cap S$ and $H^-_p\cap S$ will look near $p$ like this:

enter image description here

with the rest of the picture being topologically unaltered. Case by case analysis suggests that $H^-_p\cap S$ and $H^+_p\cap S$ cannot both be conneted: indeed, the strings can be connected 1-4;2-3 or 1-2;4-3 (1-3;2-4 would cause intersections other than $p$).