Vertex degrees of Steiner Minimal Trees in $\ell_p^d$ and other smooth Minkowski spaces

Voronoi-based estimation of Minkowski tensors from finite point samples

Intrinsic volumes and Minkowski tensors have been used to describe the geometry of real world objects. This paper presents an estimator that allows to approximate these quantities from digital images. It is based on a generalized Steiner formula for Minkowski tensors of sets of positive reach. When the resolution goes to infinity, the estimator converges to the true value if the underlying object is a set of positive reach. The underlying algorithm is based on a simple expression in terms of the cells of a Voronoi decomposition associated with the image.

The local Steiner problem in finite-dimensional normed spaces

We develop a general method of proving that certain star configurations in finit e-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner min…

Closed-Form Parametric Equation for the Minkowski Sum of $m$ Ellipsoids in $\mathbb{R}^N$ and Associated Volume Bounds

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski inequality. A reverse isometric inequality for convex bodies is also given.

广义Steiner过程的随机收敛性

Random convergence of generalized Steiner processes

本文研究广义Steiner过程的随机收敛性，建立几乎必然收敛与发散的抽象判据，并利用椭圆与三角形的联合刻画，在平面情形下完整解答了随机迭代收敛的普遍性问题。

Extremal Problems in Minkowski Space related to Minimal Networks

We solve the following problem of Z. Füredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\leq 2n and that equality holds iff the space is linearly isometric to \ell^n_\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.

随机位移的Steiner长仪与布丰偏差

Randomly Shifted Steinhaus Longimeters and Buffon Discrepancy

本文提出随机位移的Steiner长仪方法，将布丰偏差的通用上界从L^{1/3}改进到L^{1/5}(log L)^{2/5}。

关于正则单纯形的施泰纳树

On Steiner Trees of the Regular Simplex

本文重新研究高维欧几里得施泰纳树问题，建立其计算困难性与正则单纯形最优施泰纳树之间的联系，提出若干猜想，并首次给出正则单纯形最优拓扑的显式一般构造。

Diversities and the Geometry of Hypergraphs

The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into $\ell_1$. Here we show that this theory can be generalized considerably to encompass Steiner tree packing problems in both graphs and hypergraphs. Instead of the theory of $\ell_1$ metrics and minimal distortion embeddings, the parallel is the theory of diversities recently introduce…

高斯-切比雪夫求积、$\mathbb S^1$-设计与平面闵可夫斯基逆问题：设计诱导凸几何的一维模型

Gauss--Chebyshev Quadrature, $\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry

本研究探讨等权圆求积、高斯-切比雪夫求积与平面闵可夫斯基重构的关联，利用正则网格的设计性质推导求积误差、混合面积误差等的精确恒等式，得到相关误差界及扰动估计，为设计诱导凸几何提供一维模型。

Chasing Nested Convex Bodies Nearly Optimally

The convex body chasing problem, introduced by Friedman and Linial, is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep $t\in\mathbb N$, a convex body $K_t\subseteq \mathbb R^d$ is given as a request, and the player picks a point $x_t\in K_t$. The player aims to ensure that the total distance $\sum_{t=0}^{T-1}||x_t-x_{t+1}||$ is within a bounded ratio of the smallest possible offline solution. In this work, we consider the nested version of the problem, in which the sequence $(K_t)$ must be decreasing. For Euclidean spaces, we consider a memo…

关于 Minkowski 空间中 Borsuk 问题的一个注记

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。

Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions, there is a unique relationship between the position of each boundary point and the surface normal. The main results are presented as two theorems. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each surface point. Although simple to express mathematically, such a parameterization is not always practical to obtain…

Chasing Convex Bodies Optimally

In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point $x_n\in K_n$. At each step the player pays a movement cost of $||x_n-x_{n-1}||$. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial. …

Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's result is an easy consequence of the only fact that the distance function $d(x)= \dist(x,F)$ is locally DC and has no stationary point in $\R^n\setminus F$. Using this observation, we show that Ferry's (and even Fu's) result extends to sufficiently smooth normed linear spaces $X$ with…