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.

perimeterenclosing polygon. If the arc connectingaandbpassing throughcis more than 180 degrees (as it seems from the figure and from the question), then the minimum perimeter enclosing polygon would be much larger, passing on the other side. However, I do think it is the minimumareaenclosing polygon. $\endgroup$ – Yellows Jan 21 at 18:36