Search arXiv⌕ Search

arXiv · math/0208196

Non Standard Metric Products

Abstract

We consider a fairly general class of natural non standard metric products and classify those amongst them, which yield a product of certain type (for instance an inner metric space) for all possible choices of factors of this type (inner metric spaces). We further prove the additivity of the Minkowski rank for a large class of metric products.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andreas Bernig, Thomas Foertsch, Viktor Schroeder. 2002-08-26. Non Standard Metric Products. https://arxiv.org/abs/math/0208196

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Strong Negative Type of Star-shaped Manifolds

Metric spaces of negative type (NT) and strong negative type (SNT) are fundamental in wide applications across computer science and statistics. It is well-known that spheres equipped with geodesic distance are of NT, and subsets of a sphere that contain at most one pair of antipodal points (e.g. an open hemisphere) are of SNT. In this paper, we extend these results to a class of star-shaped manifolds. By constructing a metric $d$ based on layer-wise radial projections, we prove that every star-shaped manifold, equipped with $d$, is of NT. Moreover, we prove that a subset of a star-shaped manifold is of SNT under certain injectivity and directional asymmetry conditions. In particular, for a subset of a single layer, it is of SNT if it contains exactly one point from each preimage of an open hemisphere (a fully directional asymmetry); moreover, it is of SNT if and only if it contains at most one opposite-directional pair. Finally, we prove that under similar conditions, SNT holds for almost and essentially star-shaped manifolds, and for those with incomplete radial coverage.

math.MG↗

The dimension-free threshold for maximal sections of $\ell_p$

We prove the existence of a sharp threshold $p_0 \approx 26.265\ldots$, such that for every $n\in\mathbb{N}$ and $p\geq p_0$, the volume maximizing section of the unit ball of $\ell_p^n$ is the section with Ball's hyperplane $(\frac{e_1+e_2}{\sqrt{2}})^\perp$, whereas for $p\in(2,p_0)$ the asymptotic maximizer as the ambient dimension $n\to\infty$ is the hyperplane orthogonal to the diagonal vector $\frac{e_1+\cdots+e_n}{\sqrt{n}}$.

math.MG↗

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.

math.MG↗