Necessary and sufficient conditions for a set to lie in a hemisphere.
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Assume $Asubset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere).
For example, it is necessary for $A$ to not contain antipodal points. And given this condition I think it is possible to show that if it contained in a closed hemisphere then it is contained in an open one.
Furthure more, a quite strong condition - the diameter being less than $pi/2$ will suffice, although It is not necessary (think of a thin strip contained in the upper hemisphere and almost escaping it from both sides.
Another for a sufficient condition is (spherical) convexity, though of course it is not necessary.
I suspect that Borsuk-Ulam might be of help.
I ask if anyone has an idea or knows a reference? thank you!
geometry algebraic-topology
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
$endgroup$
add a comment |
$begingroup$
Assume $Asubset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere).
For example, it is necessary for $A$ to not contain antipodal points. And given this condition I think it is possible to show that if it contained in a closed hemisphere then it is contained in an open one.
Furthure more, a quite strong condition - the diameter being less than $pi/2$ will suffice, although It is not necessary (think of a thin strip contained in the upper hemisphere and almost escaping it from both sides.
Another for a sufficient condition is (spherical) convexity, though of course it is not necessary.
I suspect that Borsuk-Ulam might be of help.
I ask if anyone has an idea or knows a reference? thank you!
geometry algebraic-topology
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
$endgroup$
$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16
add a comment |
$begingroup$
Assume $Asubset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere).
For example, it is necessary for $A$ to not contain antipodal points. And given this condition I think it is possible to show that if it contained in a closed hemisphere then it is contained in an open one.
Furthure more, a quite strong condition - the diameter being less than $pi/2$ will suffice, although It is not necessary (think of a thin strip contained in the upper hemisphere and almost escaping it from both sides.
Another for a sufficient condition is (spherical) convexity, though of course it is not necessary.
I suspect that Borsuk-Ulam might be of help.
I ask if anyone has an idea or knows a reference? thank you!
geometry algebraic-topology
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
$endgroup$
Assume $Asubset S^2$ a closed subset. I am interested in necessary and sufficient conditions for $A$ to lie in a (open or closed hemisphere).
For example, it is necessary for $A$ to not contain antipodal points. And given this condition I think it is possible to show that if it contained in a closed hemisphere then it is contained in an open one.
Furthure more, a quite strong condition - the diameter being less than $pi/2$ will suffice, although It is not necessary (think of a thin strip contained in the upper hemisphere and almost escaping it from both sides.
Another for a sufficient condition is (spherical) convexity, though of course it is not necessary.
I suspect that Borsuk-Ulam might be of help.
I ask if anyone has an idea or knows a reference? thank you!
geometry algebraic-topology
geometry algebraic-topology
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
asked Dec 5 '18 at 14:27
Benny ZackBenny Zack
365213
365213
$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16
add a comment |
$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16
$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16
$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16
add a comment |
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$begingroup$
If you have two antipodal points, construct a great circle containing both. The closed hemispheres defined by these great circles both contain the two antipodal points.
$endgroup$
– ncmathsadist
Dec 5 '18 at 15:16