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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 convex hull in Riemannian world. This is Kleiner's proof of isoperimetrical inequality in 3-dimensional Hadamard space. It used the following fact:

If $K=\mathop{\rm Conv}(X)$ then Gauss curvature (= product of principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning convex hull of three point set. Generically it has ineriour points in all dimensions; see my answer here.

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:

If $K=\mathop{\rm Conv}(X)$ then the Gauss curvature (i.e. the product of the principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning the convex hull of a three point set: generically, it has interior points in all dimensions; see my answer here.

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 convex hull in Riemannian world. This is Kleiner's proof of isoperimetrical inequality in 3-dimensional Hadamard space. It used the following fact:

If $K=\mathop{\rm Conv}(X)$ then Gauss curvature (= product of principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning convex hull of three point set. Generically it has ineriour points in all dimensions; see my answer here.

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 convex hull in Riemannian world. This is Kleiner's proof of isoperimetrical inequality in 3-dimensional Hadamard space. It used the following fact:

If $K=\mathop{\rm Conv}(X)$ then Gauss curvature (= product of principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning convex hull of three point set. Generically it has ineriour points in all dimensions; see my answer here.

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 convex hull in Riemannian world. This is Kleiner's proof of isoperimetrical inequality in 3-dimensional Hadamar space. It used the following fact:

If $K=\mathop{\rm Conv}(X)$ then Gauss curvature (= product of principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning convex hull of three point set. Generically it has ineriour points in all dimensions; see my answer here.

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 convex hull in Riemannian world. This is Kleiner's proof of isoperimetrical inequality in 3-dimensional Hadamard space. It used the following fact:

If $K=\mathop{\rm Conv}(X)$ then Gauss curvature (= product of principle curvatures) of $\partial K$ at any point $p\notin X$ is zero.

Concerning convex hull of three point set. Generically it has ineriour points in all dimensions; see my answer here.