Strengthened inequalities for the mean width and the ℓ-norm

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the ℓ ‐norm of convex bodies whose Löwner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschläger verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the ℓ ‐norm of the convex hull of the support of centered isotropic measures on the unit sphere

The unrestricted blocking number in convex geometry

Let K be a convex body in \mathbb{R}^n. We say that a set of translates \left \{ K + \underline{u}_i \right \}_{i=1}^{p} block K if any other translate of K which touches K, overlaps one of K + \underline{u}_i, i = 1, . . . , p. The smallest number of non-overlapping translates (i.e. whose interiors are disjoint) of K, all of which touch K at its boundary and which block any other translate of K from touching K is called the Blocking Number of K and denote it by B(K). This thesis explores the properties of the blocking number in general but the main purpose is to study the unrestricted blocking number B_\alpha(K), i.e., when K is blocked by translates of \alpha K, where \alpha is a fixed positive number and when the restrictions that the translates are non-overlapping or touch K are removed. We call this number the Unrestricted Blocking Number and denote it by B_\alpha(K). The original motivation for blocking number is the following famous problem: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? This problem was posed by Kepler in 1611. Although this problem was raised by Kepler, it is named after Newton since Newton and Gregory had a dispute over the solution which was eventually settled in Newton’s favour. It is called the Newton Number, N(K) of K and is defined to be the maximum number of non-overlapping translates of K which can touch K at its boundary. The well-known dispute between Sir Isaac Newton and David Gregory concerning this problem, which Newton conjectured to be 12, and Gregory thought to be 13, was ended 180 years later. In 1874, the problem was solved by Hoppe in favour of Newton, i.e., N(\beta^3) = 12. In his proof, the arrangement of 12 unit balls is not unique. This is thought to explain why the problem took 180 years to solve although it is a very natural and a very simple sounding problem. As a generalization of the Newton Number to other convex bodies the blocking number was introduced by C. Zong in 1993. “Another characteristic of mathematical thought is that it can have no success where it cannot generalize.” C. S. Pierce As quoted above, in mathematics generalizations play a very important part. In this thesis we generalize the blocking number to the Unrestricted Blocking Number. Furthermore; we also define the Blocking Number with negative copies and denote it by B_(K). The blocking number not only gives rise to a wide variety of generalizations but also it has interesting observations in nature. For instance, there is a direct relation to the distribution of holes on the surface of pollen grains with the unrestricted blocking number

The blocking numbers of convex bodies

Besides determining the exact blocking numbers of cubes and balls, a conditional lower bound for the blocking numbers of convex bodies is achieved. In addition, several open problems are proposed