Vol. 22, No. 3, 1967

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On l-simplicial convexity in vector spaces

Tudor Zamfirescu

Vol. 22 (1967), No. 3, 565–573

The paper is concerned with a generalized type of convexity, which is called l-simplicial convexity. The name is derived from the simplex with l vertices, an l-simplicial convex set being the union of all (i1)-simplexes with vertices in another set, i varying between 1 and l. The basic space is a linear one.

For convex sets the l-order (which is a natural number associated to an l-simplicial convex set) is a decreasing function of l. Several inequalities between l-and k-orders are established. In doing this the case of a convex set and that of a non convex set are distinguished.

Besides these inequalities, the main result of the paper is the proof of non monotonicity of the l-order, given by an example in a 34-dimensionallinear space.

Mathematical Subject Classification
Primary: 52.30
Received: 7 March 1966
Published: 1 September 1967
Tudor Zamfirescu