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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:
           Quad on Sphere http://cs.smith.edu/~orourke/MathOverflow/HullOnSphere.jpg
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 written over a decade ago.)

Thanks for pointers and/or clarification!

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!

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:
           Quad on Sphere http://cs.smith.edu/~orourke/MathOverflow/HullOnSphere.jpg
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 written over a decade ago.)

Thanks for pointers and/or clarification!

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:
           Quad on Sphere http://cs.smith.edu/~orourke/MathOverflow/HullOnSphere.jpg
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 written over a decade ago.)

Thanks for pointers and/or clarification!