Convex and starshaped sets in manifolds without conjugate points

Sameh Shenawy

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arXiv · 1311.0454

Convex and starshaped sets in manifolds without conjugate points

Sameh Shenawy

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Abstract

Let $\mathcal{W}^{n}$ be the class of $C^{\infty }$ complete simply connected $n-$dimensional manifolds without conjugate points. The hyperbolic space as well as Euclidean space are good examples of such manifolds. Let $% W\in \mathcal{W}^{n}$ and let $A$ be a subset of $W$. This article aims at characterization and building convex and starshaped sets in this class from inside. For example, it is proven that, for a compact starshaped set, the convex kernel is the intersection of stars of extreme points only. Also, if a closed unbounded convex set $A$ does not contain a totally geodesic hypersurface and its boundary has no geodesic ray, then $A$ is the convex hull of its extreme points. This result is a refinement of the well-known Karein-Millman theorem.

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Sameh Shenawy. 2013-11-03. Convex and starshaped sets in manifolds without conjugate points. https://arxiv.org/abs/1311.0454

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