Thomas Foertsch, Viktor Schroeder
We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank only holds under additional assumptions, e.g. for Riemannian manifolds. We also study products with nonstandard product metrics.
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I apply the algebraic framework introduced in arXiv:1101.4542v3[math.MG] to Minkowski (pseudo-Euclidean) spaces in 2, 3, and 4 dimensions. The exposition follows the template established in arXiv:1307.2917[math.MG] for Euclidean spaces. The emphasis is on geometric structures, but some contact with special relativity is made by considering relativistic addition of velocities and Lorentz transformations, both of which can be seen as rotation applied to points and to lines. The language used in the paper reflects the emphasis on geometry, rather than applications to special relativity. The use o…
This paper proves that for every convex body in R^n there exist 5n-4 Minkowski symmetrizations, which transform the body into an approximate Euclidean ball. This result complements the sharp c n log n upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean ball.
It is surprising, but an established fact that the field of Elementary Geometry referring to normed spaces (= Minkowski spaces) is not a systematically developed discipline. There are many natural notions and problems of elementary and classical geometry that were never investigated in this more general framework, although their Euclidean subcases are well known and this extended viewpoint is promising. An example is the geometry of simplices in non-Euclidean normed spaces; not many papers in this direction exist. Inspired by this lack of natural results on Minkowskian simplices, we present a …
A synthetic proof of the spherical and hyperbolic Pythagorean theorem on models in Euclidean and Minkowski space
针对球面与双曲几何中涉及圆盘面积的毕达哥拉斯定理推广形式尚无综合证明的问题，本文借助欧氏空间球面和闵氏空间双曲面模型，给出了该定理的初等几何综合证明。
There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We present new generalizations of well known properties of Euclidean simplices. These results refer to analogues of circumcenters, Euler lines, and Feuerbach spheres of simplices in normed spaces. Using duality…
On the (outer) Minkowski content with lower-dimensional structuring element
本文研究了带有低维结构元素的外Minkowski内容，探讨了其存在性条件及在三维空间中的应用示例。
For a convex set (K) of the (n)-dimensional Euclidean space, the Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional Euclidean volume of the neighborhood of the radius (t). Being defined for positive (t), the Steiner-Minkowski polynomials are considered for all complex (t). The renormalization procedure for Steiner polynomial is proposed. The renormalized Steiner-Minkowski polynomials corresponding to all possible solid convex sets in all dimensions form a normal family in the whole complex plane. For each of the four families of convex sets: the Euclidean balls, the cubes,…
It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also invariant with respect to the ambient space when normalized by an appropriate constant. In other words, the value of the normalized Minkowski content of a bounded, Minkowski measurable set is intrinsic to the set.
We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minko…
Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a symmetric convex body $\mathfrak x$, we try to maximize the volume of $\mathfrak x$ and minimize the width of $\mathfrak x$ simultaneously.
It is well known that in $n$-dimensional Euclidean space ($n\geq 2$) the classes of (diametrically) complete sets and of bodies of constant width coincide. Due to this, they both form a proper subfamily of the class of reduced bodies. For $n$-dimensional Minkowski spaces, this coincidence is no longer true if $n\geq 3$. Thus, the question occurs whether for $n\geq 3$ any complete set is reduced. Answering this in the negative for $n\geq 3$, we construct $(2^{k}-1)$-dimensional ($k\geq 2$) complete sets which are not reduced.
The structure of $k$-diametral point configurations in Minkowski $d$-space is shown to be closely related to the properties of $k$-antipodal point configurations in $\mathbb{R}^d$. In particular, the maximum size of $k$-diametral point configurations of Minkowski $d$-spaces is obtained for given $k\geq 2$ and $d\geq 2$ generalizing Petty's results (Proc. Am. Math. Soc. 29: 369-374, 1971) on equilateral sets in Minkowski spaces. Furthermore, bounds are derived for the maximum size of $k$-diametral point configurations in Euclidean $d$-space. In the proofs convexity methods are combined with vol…
A generalization of the Minkowski distance and a new definition of the ellipse
本文提出n维空间中Minkowski距离的一种推广，证明其构成度量族，研究三种特殊情形的圆的性质，并由此给出椭圆的新定义。
In this paper we study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth centrally symmetric and strictly convex body $K$, for every convex caustic which $K$ possesses, the "dual" billiard dynamics in which the table is the Euclidean unit disk and the geometry that governs the motion is induced by the body $K$, possesses a dual convex caustic. Such a pair of caustics is dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation num…
Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
本文利用相对分形鼓的分形 ζ 函数及其主复维数，建立任意维欧几里得空间中 RFD 的 Minkowski 可测性准则，并推广分形弦的经典结论。
Let $Λ$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $Λ'$ be a Minkowskian sublattice of $Λ$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $Λ$. We extend the classification of possible $\Z/d\Z$-codes of the quotients $Λ/Λ'$ to dimension~$9$, where $d\Z$ is the annihilator of $Λ/Λ'$.
A Minkowski problem for $α$-concave functions via optimal transport
本文通过总质量泛函的第一变分引入$α$-凹函数的表面积测度概念，将其推广至$α$-凹测度，并利用最优传输研究相关的Minkowski问题。
Minkowski weak embedding theorem
本研究针对有限闵可夫斯基维数空间，证明了Assouad嵌入定理的非单射类比结论，实现非加倍空间的弱嵌入，为随机几何、数学物理领域的相关空间研究提供了新工具。
It is a classical fact, that given an arbitrary n-dimensional convex body, there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis, our technique involves spherical harmonics. We also make use o…
A note on Borsuk's problem in Minkowski spaces
本文研究 Minkowski 空间中的 Borsuk 问题，指出 Kahn–Kalai 的反例构造可推广至所有 $\ell_p$ 空间，且其划分下界随 $p$ 增大而增强。
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