Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1

In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

A note on a three-dimensional sphere theorem with integral curvature condition

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.