Based on the definition of divisibility of Markovian quantum dynamics, we discuss the Markovianity of tensor products, multiplications and some convex combinations of Markovian quantum dynamics. We prove that the tensor product of two Markovian dynamics is also a Markovian dynamics and propose a new witness of non-Markovianity.

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

Consider a collection of topological spheres in Euclidean space whose intersections are essentially topological spheres. We find a bound for the number of components of the complement of their union and discuss conditions for the bound to be achieved. This is used to give a necessary condition for independence of these sets. A related conjecture of Griinbaum on compact convex sets is discussed.

First we construct spreads consisting of analytic curves (circular arcs and segments), without points of finite multiplicity. Then we see that, in the sense of Baire categories, most such spreads have no points of finite multiplicity.