One of the fascinating features of the theory of convex bodies is the wealth of substantial
results that spring from the mere assumption of convexity. The present survey is concerned …

When S. Kobayashi wrote his chapter for the first edition of this book he reported on amazing
results which had just been obtained. Those results triggered a rapid development of …

Let M be a connected, complete Riemannian manifold with (possibly empty) boundary dM.
Cheeger and Gromoll proved in [4] that if a1V1 is empty and the Ricci curvature of M is …

There are reported results on the following problems:(i) on the validity of the Penrose
inequality;(ii) on the hoop conjecture;(iii) on the physical criteria for the formation of trapped …

The broad subject of this dissertation can be described as “the geometry of the second
fundamental form.” I have not restricted myself to a mere presentation of the handful new …

Question:" Under what curvature assumptions on a complete open manifold is the volume of
balls of a fixed radius bounded below independent of the center point?" Two theorems …

In this paper we describe the topology of 4-dimensional closed orientable Riemannian
manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound …

Rauch [41] proved a fundamental theorem on the lengths of Jacobi fields, called the Rauch
comparison theorem. After him, Berger [4], Warner [48] and Heintze and Karcher [27] etc …

For Riemannian manifolds V9 V we define dev (F, V) to be the" inf" of the deviations of all
diffeomorphisms V-* V. We treat" dev" as a metric in the set of isometry classes of …

In this paper, we present sharp stability results for various reverse isoperimetric problems in
$\mathbb R^ 2$. Specifically, we prove the stability of the reverse isoperimetric inequality for …