Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0603616

The local Steiner problem in finite-dimensional normed spaces

Konrad J Swanepoel
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 minimal tree in this norm. The proof makes essential use of extremal finite set theory. The second norm, occuring in the work of Mark Conger (1989), is the sum of the ell_1-norm and a small multiple of the ell_2 norm. For the second norm we determine the maximum degree of a Steiner point.

Explore related subjects

math.MGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Konrad J Swanepoel. 2006-07-19. The local Steiner problem in finite-dimensional normed spaces. https://doi.org/10.1007/s00454-006-1298-z

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

KEEP EXPLORING

Related papers

arXiv2026

From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization

For a triangle of area \(T\), a planar corner-area characterization states that an interior point \(M\) lies on the Steiner inellipse precisely when the three corner triangles cut off by the lines through \(M\) parallel to the sides have areas \(T_1,T_2,T_3\) satisfying $$ T_1+T_2+T_3=\frac12 T. $$ We give the corresponding statement for a simplex in arbitrary dimension. If \(S\) is a nondegenerate \(n\)-simplex of volume \(V\) and \(V_1(M),\ldots,V_{n+1}(M)\) are the volumes of the facet-parallel corner simplices determined by \(M\), then $$ M\in\partial E_J(S) \quad\Longleftrightarrow\quad \sum_{i=1}^{n+1}V_i(M)^{2/n}=\frac1n V^{2/n}, $$ where \(E_J(S)\) is the John ellipsoid of \(S\). We also identify the entire corner-volume functional with the central second-moment quadratic of the uniform simplex. The novelty claimed here is limited to the corner-volume formulations and their connections with the planar Steiner-inellipse result; the underlying barycentric, covariance, and John-ellipsoid facts are classical.

math.MG↗
arXiv2026

The $L_p$ Minkowski problem for $C$-close sets: existence and continuity

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $μ$ defined on $Ω_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $μ$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $μ$ being a nonzero finite Borel measure on $Ω_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

math.MG↗
arXiv2026

Convergence of metric measure spaces via embeddings in the Urysohn universal space

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $μ\simν$ if $ν$ is the pushforward of $μ$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.

math.MG↗