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Distance on the sphere is convex

0

$begingroup$

Let $mathbbS^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:mathbbS^n to [0, infty)$ defined by $d(p) = d(p,p_0) = cos^-1( langle p, p_0 rangle)$. It is the intrinsec distance to $p_0$ in the sphere. Is this function convex?

In order to answer this question, we have to show that $d circ gamma$ is a convex function of a real variable, for any geodesic $gamma$ of the sphere. I showed it for geodesics issuing from $p_0$. Is it enough?

smooth-manifolds spheres spherical-geometry geodesic convex-geometry

share|cite|improve this question

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

$endgroup$

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  Sorry. Geodesics through $p_0$ tell you only how the function behaves as you move away from $p_0$. They do not tell you how it behaves in other directions.
  $endgroup$
  – Paul Sinclair
  Apr 3 at 0:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. Do you believe that $d$ is convex?
  $endgroup$
  – Eduardo Longa
  Apr 3 at 1:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look at $Bbb S^2$ first. Consider a great circle slanted at 45 degrees. Calculate your function as a function of $theta$ around the circle. Is that function convex everywhere?
  $endgroup$
  – Paul Sinclair
  Apr 3 at 1:13
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $mathbbS^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:mathbbS^n to [0, infty)$ defined by $d(p) = d(p,p_0) = cos^-1( langle p, p_0 rangle)$. It is the intrinsec distance to $p_0$ in the sphere. Is this function convex?

In order to answer this question, we have to show that $d circ gamma$ is a convex function of a real variable, for any geodesic $gamma$ of the sphere. I showed it for geodesics issuing from $p_0$. Is it enough?

smooth-manifolds spheres spherical-geometry geodesic convex-geometry

share|cite|improve this question

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry. Geodesics through $p_0$ tell you only how the function behaves as you move away from $p_0$. They do not tell you how it behaves in other directions.
  $endgroup$
  – Paul Sinclair
  Apr 3 at 0:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you for your comment. Do you believe that $d$ is convex?
  $endgroup$
  – Eduardo Longa
  Apr 3 at 1:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look at $Bbb S^2$ first. Consider a great circle slanted at 45 degrees. Calculate your function as a function of $theta$ around the circle. Is that function convex everywhere?
  $endgroup$
  – Paul Sinclair
  Apr 3 at 1:13
  

  
  
  
  

add a comment | 

$begingroup$

Let $mathbbS^n$ be the unit sphere and choose $p_0$ as the north pole. Consider the function $d:mathbbS^n to [0, infty)$ defined by $d(p) = d(p,p_0) = cos^-1( langle p, p_0 rangle)$. It is the intrinsec distance to $p_0$ in the sphere. Is this function convex?

In order to answer this question, we have to show that $d circ gamma$ is a convex function of a real variable, for any geodesic $gamma$ of the sphere. I showed it for geodesics issuing from $p_0$. Is it enough?

smooth-manifolds spheres spherical-geometry geodesic convex-geometry

share|cite|improve this question

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

$endgroup$

share|cite|improve this question

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

share|cite|improve this question

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719

asked Apr 2 at 17:35

Eduardo LongaEduardo Longa

1,8952719