Simplexes in spaces of constant curvature

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form _n (l)l _ijl which guarantees the existence of an n-simplex p ₁ p ₂ ... p _n+1 with edge-lengths p _ip_j=l _ij. (In spherical geometry of curvature 1 the interval is empty unless l2 arcsin ) The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.This work was supported by Canadian NSERC grants.     

On convex bodies of constant width

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.     

Geodesics in Randers spaces of constant curvature

Geodesics in Randers spaces of constant curvature are classified.

     

Incidence theorems in spaces of constant curvature

Certain analogs of the classic theorems of Menelaus and Ceva are considered for a hyperbolic surface, a sphere, and for three-dimensional hyperbolic and spherical spaces.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 67–74, 1992.     

Spherical means in spaces of constant curvature

Summary Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations. To Enrico Bompiani on his scientific Jubiles This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under ontract No. AF 49 (638)–228.     

Integral geometry in constant curvature Lorentz spaces

We extend classical Cauchy formulas and Crofton formulas to the Lorentz-Minkowski space, and to constant curvature Lorentz spaces.     

Forced convex mean curvature flow in Euclidean spaces

We show the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.     

On ricci-pseudosymmetric hypersurfaces in spaces of constant curvature

Results in Mathematics - This article studies the conditions of pseudosymmetry and Ricci-pseudosymmetry, realizedon hypersurfaces of semi-Riemannian spaces of constant curvature. In particular, we...     

Radon transform on spaces of constant curvature

A correspondence among the totally geodesic Radon transforms-as well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.

     

Functionals on the spaces of convex bodies

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.