The article deals with the relationship between Laguerre geometry and Minkowski sum. This relationship is used to create new geometry based generally on fundamental objects different to closed balls that formed Laguerre geometry. In the end of the article, there is defined also generalization of medial axis transform to the newly defined spaces and proved its uniqueness. The article continues t...

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g ≥ 1 2 cosh(r) where r denotes the radius of any isometrically embedded ball in M . Assuming an unpublished result of Pitts and Rubinstein improves this to g ≥ 1 2 cosh(r) + 1 2 . We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperim...

The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained. © 2006 Elsevier Inc. All rights reserved.

Forty years ago Schaer and Wetzel showed that a $\frac{1}{\pi}\times\frac {1}{2\pi}\sqrt{\pi^{2}-4}$ rectangle, whose area is about $0.122\,74,$ is the smallest rectangle that is a cover for the family of all closed unit arcs. More recently F\"{u}redi and Wetzel showed that one corner of this rectangle can be clipped to form a pentagonal cover having area $0.11224$ for this family of curves. He...

An algorithm is presented which, given the parameters of an Iterated Function System (IFS) which uses affine maps, constructs a closed ball which completely contains the attractor set of the IFS. These bounding balls are almost always smaller than those computed by existing methods, and are sometimes much smaller. The algorithm is numerical in form, involving the optimisation of centre-point an...

Using a novel Wintner-type formulation of the classical Peano’s existence theorem [Math. Ann. 37 (1890), 182-228], we enhance Ważewski’s result on invertibility of maps defined on closed balls [Ann. Soc. Pol. Math. 20 (1947), 81-125] securing the size of the domain of invertibility that agrees with the bounds derived by John [Comm. Pure Appl. Math. 21 (1968), 77-110] and Sotomayor [Z. Angew. Ma...

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g ≥ 1 2 cosh(r) where r denotes the radius of any isometrically embedded ball in M . Assuming an unpublished result of Pitts and Rubinstein improves this to g ≥ 1 2 cosh(r) + 1 2 . We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperim...

Let Γ be a lattice in G = SL(n,R) and X = G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of Γ in X analogous to the classical equidistribution on torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of Borel subgroup ...

We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing distance inequalities. also a sharp upper estimate outer volume ratio from an arbitrary convex body unit balls subspaces $L_p$.

Two Riemannian manifolds are said to be isotonal if the elements of the spectra are the same, but the multiplicities may be different. In this lecture we describe general non-continuous so-called σ-deformations which provide isospectral manifolds (closed as well as manifolds with boundaries) with different local geometries. These spaces have also the following spectral-geometric coincidences: (...