Can a Jordan convex curve be rewritten as the image of 4 monotone real functions?
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Is it true that a smooth Jordan curve $C subseteq mathbb{R}^2$ that is convex (in the sense that the region bounded by this curve is convex) can be rewritten as the union of the image sets of 4 monotone functions $phi_j colon I_j subseteq mathbb{R} mapsto mathbb{R}$ ?
If $C$ is a circle or an ellipse then it is true as we have explicit parametrizations. Considering this I would like to know what happens in a more general context.
geometry differential-geometry convex-analysis plane-curves convex-geometry
asked Nov 29 '18 at 15:03
LAELAE
112
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add a comment |
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Is it true that a smooth Jordan curve $C subseteq mathbb{R}^2$ that is convex (in the sense that the region bounded by this curve is convex) can be rewritten as the union of the image sets of 4 monotone functions $phi_j colon I_j subseteq mathbb{R} mapsto mathbb{R}$ ?
If $C$ is a circle or an ellipse then it is true as we have explicit parametrizations. Considering this I would like to know what happens in a more general context.
geometry differential-geometry convex-analysis plane-curves convex-geometry
asked Nov 29 '18 at 15:03
LAELAE
112
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A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
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– Michael Hoppe
Nov 29 '18 at 15:09
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Yes. Consider the minimum/maximum of both coordinates of points in $C$.
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– Berci
Nov 29 '18 at 15:10
add a comment |
$begingroup$
Is it true that a smooth Jordan curve $C subseteq mathbb{R}^2$ that is convex (in the sense that the region bounded by this curve is convex) can be rewritten as the union of the image sets of 4 monotone functions $phi_j colon I_j subseteq mathbb{R} mapsto mathbb{R}$ ?
If $C$ is a circle or an ellipse then it is true as we have explicit parametrizations. Considering this I would like to know what happens in a more general context.
geometry differential-geometry convex-analysis plane-curves convex-geometry
asked Nov 29 '18 at 15:03
LAELAE
112
$endgroup$
Is it true that a smooth Jordan curve $C subseteq mathbb{R}^2$ that is convex (in the sense that the region bounded by this curve is convex) can be rewritten as the union of the image sets of 4 monotone functions $phi_j colon I_j subseteq mathbb{R} mapsto mathbb{R}$ ?
If $C$ is a circle or an ellipse then it is true as we have explicit parametrizations. Considering this I would like to know what happens in a more general context.
geometry differential-geometry convex-analysis plane-curves convex-geometry
geometry differential-geometry convex-analysis plane-curves convex-geometry
asked Nov 29 '18 at 15:03
LAELAE
112
asked Nov 29 '18 at 15:03
LAELAE
112
asked Nov 29 '18 at 15:03
LAELAE
112
asked Nov 29 '18 at 15:03
LAELAE
112
asked Nov 29 '18 at 15:03
LAELAE
112
112
$begingroup$
A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
$endgroup$
– Michael Hoppe
Nov 29 '18 at 15:09
$begingroup$
Yes. Consider the minimum/maximum of both coordinates of points in $C$.
$endgroup$
– Berci
Nov 29 '18 at 15:10
add a comment |
$begingroup$
A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
$endgroup$
– Michael Hoppe
Nov 29 '18 at 15:09
$begingroup$
Yes. Consider the minimum/maximum of both coordinates of points in $C$.
$endgroup$
– Berci
Nov 29 '18 at 15:10
$begingroup$
A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
$endgroup$
– Michael Hoppe
Nov 29 '18 at 15:09
$begingroup$
A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
$endgroup$
– Michael Hoppe
Nov 29 '18 at 15:09
$begingroup$
Yes. Consider the minimum/maximum of both coordinates of points in $C$.
$endgroup$
– Berci
Nov 29 '18 at 15:10
$begingroup$
Yes. Consider the minimum/maximum of both coordinates of points in $C$.
$endgroup$
– Berci
Nov 29 '18 at 15:10
add a comment |
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$begingroup$
A well known problem, not easy to solve, see en.wikipedia.org/wiki/Four-vertex_theorem.
$endgroup$
– Michael Hoppe
Nov 29 '18 at 15:09
$begingroup$
Yes. Consider the minimum/maximum of both coordinates of points in $C$.
$endgroup$
– Berci
Nov 29 '18 at 15:10