Some results on the geometry of convex hulls in manifolds of pinched negative curvature (Q1327473)

Let \(X\) be a complete simply-connected Riemannian manifold whose curvature lies between two negative constants, let \(X_I\) be its ideal boundary and let \(X_C=X\cup X_I\). Recall that between two distinct points \(x\) and \(y\) of \(X_C\), there is a unique geodesic \([x,y]\), and a subset \(A\subset X_C\) is called convex if for all \(x\) and \(y\) in \(A\), we have \([x,y]\subset A\). The convex hull hull\((Q)\) of a closed set \(Q\subset X\) is the intersection of all the closed convex sets containing \(Q\). The paper develops general properties of convex hulls in \(X\) and \(X_C\). The first result is the following: Let \(\mathcal C(X_C)\) be the set of all closed subsets of \(X_C\), equipped with the Hausdorff topology. Then the map \(Q\mapsto \text{hull}(Q)\) defined from \(\mathcal C(X_C)\) to itself is continuous. In fact, there are estimates using a path metric which the author puts on \(X_C\) generalizing a construction of Floyd. The author then describes a general theory of approximating finite sets of points by trees, which is a refinement of a construction of Gromov. He proves that the convex hull of a finite set of points is, in a certain precise sense, ``treelike''. Then he studies volumes of convex hulls of finite sets of points and of tubular neighborhoods of geodesics. The methods use in particular a variation of a construction of convex sets due to M. Anderson. One of the results is that the volume of convex hulls of \(n\) points is a continuous function in these points, provided no two of them converge to the same ideal point. This volume is bounded by a constant depending only on \(n\), on the dimension of \(X\) and on the pinching constants for the curvature.