We study the relative equilibria of the limit case of the planar Newtonian 4–body problem when three masses tend to zero, the so-called (1 + 3)–body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1 + ...

In the following, 5 will denote the boundary of the unit ball in En, and u a variable point of S (so re is a unit vector, or "direction"). The polar equation of the boundary of DK is given by p = /»(«), uES, so piu) is the radius of DK in the direction u. Then p(w) is the maximum length of a chord of K having direction u—the length of a "diameter" of K having direction u. Let p. denote w-dimens...

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

New results on approximation of a convex body K ⊂ R3 by affine images of circular cylinders, parallelepipeds, hexagonal and octagonal regular (and some other) prisms are obtained. Two of the theorems obtained are as follows (V (K) denotes the volume of a body K ⊂ R3). Theorem 1. Let K be an arbitrary convex body in R3. There exists a regular octagonal prism an affine image of which is circumscr...

It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent points with this property.

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...