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.)
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