We investigate the following two problems the first of which is on sphere packings in Euclidean space and the second of which is closely connected to sphere coverings in spherical space. Thus, first

we a raise a relative of the well-known Kelvin problem that one can regard as a stronger version of the Kepler conjecture. In particular, we prove that the average surface area of any family of convex cells that tile the Euclidean 3-space with each cell containing a unit ball, is always at least 13.8564... . Second recall that a subset of the d-dimensional Euclidean space having nonempty interior is called a

spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a

``fat'' one, if it contains the centers of its generating balls. We outline a proof of the X-ray Conjecture for fat spindle convex bodies in dimensions 3, 4, 5, and 6 as well as in dimensions greater than or equal to 15.