Are shortest halving curves simple closed geodesics? 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.
 A: Look at it from a calculus of variations perspective. A geodesic is locally a distance-minimizing curve - that is, an infinitesimal change in the curve between two points, which keeps those two points constant, cannot decrease the length.
Your curve is globally a distance minimizing curve among curves with a fixed area on one side. Using Lagrange multipliers, we can see that the curve minimizes the sum of the length plus some constant multiple of the area on one side. If that constant multiple is zero, you obtain a geodesic. If not, you don't.
Solving the minimization problem for a given constant gives a differential equation. I haven't done the computation, but I would guess that, if a geoedesic represents travelling in a straight line on the surface, then this equation represents travelling with a constant rate of turning.
A: 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):

 
 
 


