A variational inequality for the images of k-dimensional hyper-planes under quasiconformal maps of the n-dimensional Euclidean space is proved when 1 ≤ k ≤ n − 2 .

A finite set H in Rd is called an acute set if any angle determined by three points of H is acute. We examine the maximal cardinality α(d) of a d-dimensional acute set. The exact value of α(d) is known only for d ≤ 3. For each d ≥ 4 we improve on the best known lower bound for α(d). We present different approaches. On one hand, we give a probabilistic proof that α(d) > c · 1.2d. (This improves ...

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces whic...

In this paper, vision theory for Euclidean, spherical and hyperbolic spaces is studied in a uniform framework using diierential geometry in spaces of constant curvature. It is shown that the epipolar geometry for Euclidean space can be naturally generalized to the spaces of constant curvature. In particular, it is shown that, in the general case, the bilinear epipolar constraint is exactly the ...

In this paper we prove some results on the computational complexity of standard quantifierfree spatial logics with the connectedness predicate interpreted over the Euclidean spaces R and R. Topological logics with connectedness. A topological logic is a formal language whose variables range over subsets of topological spaces, and whose non-logical primitives denote fixed topological properties ...

A set of lines through the origin is called equiangular if every pair of lines defines the same angle, and the maximum size of an equiangular set of lines in R was studied extensively for the last 70 years. In this paper, we study analogous questions for k-dimensional subspaces. We discuss natural ways of defining the angle between k-dimensional subspaces and correspondingly study the maximum s...

B.Y. Chen introduced biharmonic submanifolds in Euclidean spaces and raised the conjecture ”Any biharmonic submanifold is minimal”. In this article, we show some affirmative partial answers of generalized Chen’s conjecture. Especially, we show that the triharmonic hypersurfaces with constant mean curvature are minimal. M.S.C. 2010: 58E20, 53C43.