On Maximal Primitive Fixing Systems

Let max (M) denote the maximal number of points forming a primitive xing system for a convex body M R n. Sharpening results of B. Bollobb as with respect to the quantity max (M) for n 3, we construct counterexamples to the conjecture max (M) 2(2 n ? 1) of L. Danzer. These counterexamples M satisfy max (M) = 1, and they can belong to the following classes of convex bodies: cap bodies, zonoids, and zonotopes.