arXiv subjects
Publications and source records attributed to Olaf Mordhorst.
This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $s \nearrow 1$ is proven. It is shown that their limit as $s \searrow -\infty$ can be expressed in terms of the volume.
We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.
The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.
We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.
We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of second category. We investigate extremal cases for a class of symmetry measures. We show that the center of the John and Löwner ellipsoid can be far apart and we give the optimal order for the extremal distance of the two centers.
A mean step in Haagerup's proof for the optimal constants in Khintchine's inequality is to show integral inequalities of type $\int(g^s-f^s)\mathrm{d}μ\geq 0$. F.L. Nazarov and A.N. Podkorytov made Haagerup's proof much more clearer for the case 0<p<2 by using a lemma on distribution functions. In this article we want to treat the case 2<p<3 with their technique.