On Maximal Primitive Fixing Systems

. Let ae max (M) denote the maximal number of points forming a primitive fixing system for a convex body M ae R n . Sharpening results of B. Bollob'as with respect to the quantity ae max (M) for n 3, we construct counterexamples to the conjecture ae max (M) 2(2 n \Gamma 1) of L. Danzer. These counterexamples M satisfy ae max (M) = 1, and they can belong to the following classes of convex bodies: cap bodies, zonoids, and zonotopes. MSC 1991: 52A20 Keywords: convex body, illumination, primitive fixing system, cap body, zonoid, zonotope, Danzer conjecture. 1. Introduction L. Fejes T'oth [7] introduced the notion of a primitive fixing system for a compact, convex body in R n . We recall the definition and some related results. Let M ae R n be a compact, convex body and F ae bd M . A direction l, defined by a nonzero vector e 2 R n , is said to be an outer moving direction with respect to M and F if for every real number ? 0 the intersection of F and (\Gammae+ int M) is empty..