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.
Best Answer
Q1. For sure simply connected complete and curvature $\le 0$ is sufficient. It is also true for any complete metric on $\mathbb R^2$ without conjugate points.
Q2. Almost no properties survive. I saw only one application of the convex hull in the Riemannian world. This is Kleiner's proof of the isoperimetric inequality in 3-dimensional Hadamard space. It used the following fact:
Concerning the convex hull of a three point set: generically, it has interior points in all dimensions; see my answer here.