ACTA issues

On non-onesided $M$-complete vector systems

Vladimir Boltyanski, Horst Martini

Acta Sci. Math. (Szeged) 74:1-2(2008), 297-313

Abstract. The notion of $H$-convexity is a generalized convexity notion with many metrical and combinatorial applications (e.g., in distance geometry, combinatorial geometry, Minkowski geometry, and abstract convexity), and $H$-convex sets are simply defined with the help of a finite or infinite system $H$ of unit vectors in Euclidean $n$-space. In [BM2], [BM3], and [BM4] we investigated non-onesided, so-called $M$-complete systems of unit vectors and some of their applications in combinatorial geometry. In particular, we established a condition under which the vector (or Minkowski) sum of any two $H$-convex sets is again $H$-convex, and conditions for $H$-separability of $H$-convex sets. In both cases the notion of $M$-completeness, defined for the vector systems $H$, plays the key role. Here we study properties of {\it maximal} non-onesided, $M$-complete vector systems $\overline H$ and $\hat H$ in the unit sphere ${\msbm S}^{n-1}$, which means that any non-onesided, $M$-complete vector system containing them coincides with ${\msbm S}^{n-1}$. On the other hand, we prove for closed systems, which are symmetric with respect to the origin, that the systems $\overline H$ and $\hat H$ are also {\it universal}, i.e., under some natural condition every non-onesided, $M$-complete vector system distinct from ${\msbm S}^{n-1}$ is contained in $\overline H$ or in $\hat H$. Some examples illustrate the obtained results.

AMS Subject Classification (1991): 32F17, 32F99, 52A01, 52A20, 52A30

Keyword(s): direct decomposition, direct vector sum, generalized convexity notion, H, -convexity, M, -complete vector system, Minkowski addition, positive linear combination, universality, vector sum

Received May 23, 2007, and in revised form October 1, 2007. (Registered under 6/2007.)