TITLE: Are shortest halving curves simple closed geodesics?
QUESTION [6 upvotes]: Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a shortest halving curve if
(a) it partitions the surface area of $S$ into two equal-area halves,
and (b) it is the shortest such curve (under the Euclidean metric).

 



Q. Is $\gamma$ necessarily a simple (non-self-intersecting) closed geodesic on $S$? 

On a polyhedral surface, the equivalent would be a simple closed quasigeodesic
(in Alexandrov's sense).
 Addendum.
Douglas Zare's counterexample to that vain hope:



 
 
 
 
 
 
 
 
 
 
 
(Not metrically accurate.)

The red circle seems to be a shortest halving curve, but it is not a geodesic.

REPLY [3 votes]: I just located this relevant reference:

Engelstein, Max, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard. "Isoperimetric problems on the sphere and on surfaces with density."
  New York J. Math 15 (2009): 97-123. (PDF download link)
"the short equator gives a least perimeter partition of the ellipsoid into
  two regions of equal area"


 


Note the similarity in Fig.7 below to Will Sawin's disc-bubble
(although making a different point):