arXiv subjects
Publications and source records attributed to Hauke Dirksen.
We compute the best constant in the Khintchine inequality for equidistributed random variables on the $N$-sphere in the Orlicz space $L_{ψ_2}$.
We state a general formula to compute the volume of the intersection of the regular $n$-simplex with some $k$-dimensional subspace. It is known that for central hyperplanes the one through the centroid containing $n-1$ vertices gives the maximal volume. We show that, for fixed small distances of a hyperplane to the centroid, the hyperplane containing $n-1$ vertices is still volume maximizing. The proof also yields a new and short argument for the result on central sections. With the same technique we give a partial result for the minimal central hyperplane section. Finally, we obtain a bound for $k$-dimensional sections.
We provide a formula to compute the volume of the intersection of a generalized cylinder with a hyperplane. Then we prove an integral inequality involving Bessel functions similar to Keith Ball's well-known inequality. Using this inequality we obtain upper bounds for the section volume. For large radius of the cylinder we determine the maximal section.