TITLE: Convex hull on a Riemannian manifold
QUESTION [15 upvotes]: Let $M$ be a complete Riemannian 2-manifold.
Define a subset $C$ of $M$ to be convex
if all shortest paths between any two points
$x,y \in C$ are completely contained within $C$.
For a finite set of points $P$ on $M$, define
the convex hull of $P$ to be
the intersection of all convex sets containing $P$.
It is my understanding that this definition is due to Menger.
In the Euclidean plane, the convex hull of $P$ coincides
with the minimum perimeter polygon enclosing $P$.
This does not hold on every $M$.
For example, the convex hull of four points on a sphere that do not
fit in a hemisphere is the whole sphere (this is Lemma 3.4 in the book below),
different from the minimum perimeter geodesic polygon:

          


The shortest path connecting $a$ and $b$ goes around the back of the sphere,
but the illustrated quadrilateral is (I think!) the minimum perimeter polygon enclosing 
$\lbrace a,b,c,d \rbrace$.
My specific question is:

Q1.
  Under what conditions on $M$ and on $P$ will the convex hull of $P$
  coincide with the minimum perimeter geodesic polygon enclosing $P$?

I am teaching the (conventional, Euclidean)
convex hull now, and it would be enlightening to say something
about generalizing the concept to 2-manifolds.
More generally:

Q2.
  Which properties of the convex hull in $\mathbb{R}^d$ are retained
  and which lost when generalizing to the convex hull in a
  $d$-manifold?

(The earlier MO question,
Convex Hull in CAT(0),
is related but its focus is different.)
I recall reading somewhere in Marcel Berger's writings that some questions about
convex hulls of just three points in dimension $d > 3$ are open, but I cannot find the
passage at the moment, and perhaps he was discussing a different concept of hull...
Added: I found the passage, in Berger's
Riemannian geometry during the second half of the twentieth century (American Mathematical Society, Providence, 2000), p.127:

A most naive problem is the following.  What is the convex envelope of $k$ points in a Riemannian
  manifold of dimension $d \ge 3$?  Even for three points and $d \ge 3$ the question is
  completely open (except when the curvature is constant).  A natural example to look at would be
  $\mathbb{C P}^2$, because it is symmetric but not of constant curvature.

(Caveat: These quoted sentences were published in 2000.)
Thanks for pointers and/or clarification!
C. Grima and A. Márquez,

Computational Geometry on Surfaces:
Performing Computational Geometry on the Cylinder, the Sphere, the Torus, and the Cone,
Springer, 2002.

REPLY [6 votes]: Convex hulls are extremely useful in hyperbolic/negatively curved geometry (in particular, I disagree with Anton's bold accessment in his answer that convex hulls have few applications in Riemannian world). Search on the phrases "convex core" and "hyperbolic manifold" to see how convex hulls help to work with Kleinian groups and (non-simply-connected) hyperbolic manifolds.
In the simply-connected case there is the following basic fact: if two convex sets $A$, $B$ in the hyperbolic $n$-space intersect, then the convex hull of $A\cup B$ is contained in the $\Delta$-neighborhood of $A\cup B$, where $\Delta$ is the thin triangle constant for the hyperbolic plane. (See e.g. Lemma 2.12 of this paper by Baker-Cooper). This illustrates fundamental difference of hyperbolic and Euclidean geometry.
One place where convex hulls appear prominently is the paper "The Dirichlet problem at infinity for manifolds of negative curvature" by Michael Anderson, JDG (1983), 701-721.
My personal favorite application of this work (derived by Bowditch) is that in a simply-connected manifold of pinched negative curvature the convex hull of any quasi-convex set $Q$ is within bounded Hausdorff distance of $Q$. Surprisingly, this result requres that the curvature is pinched negative and not merely negative.