on the Steiner tree
connecting
a fractal set

[E. Paolini: Università di Pisa]

Brixen ‐ Jun 2, 2023


Joint work with E. Stepanov

Plan

  • the classical Steiner Problem
  • generalization to infinite sets
  • minimality of the infinite self-similar complete binary tree

Fermàt point

(Cavalieri, Torricelli)

Extension proposed by Steiner


Generalization proposed by Courant-Robbins (1941)


Formalization given by Jarník-Kössler (1934)

Steps for existence:

  • no loops
  • a priori bound on topology
  • reduction to finite dimensional problem

Examples:

Melzak's construction

Classical approach: union of curves,
Ivanov-Tuzhilin (1984)

Set theoretic approach: compact sets, Hausdorff measure Ambrosio-Tilli (2004)

GMT approach: $G$-currents and calibrations, Marchese-Massacesi (2016)

Set theoretic approach

  • the ambient is a metric space $X$
  • the given is a compact set $\color{yellow} A$
  • the unknown is a set $\color{white}S$
  • the energy is the Hausdorff $\mathcal H^1$ measure
  • the constraint is topological connectedness

Tools

  • compactness: the family of compact subsets of a compact space is compact in the Hausdorff metric (Blaschke)
  • lower-semicontinuity: $\mathcal H^1$ is l.s.c. with respect to the Hausdorff metric on connected sets (Gołąb)

possible formulations

Given a compact set $\color{yellow} A$, find $\color{white} S$ such that $\mathcal H^1({\color{white}S})$ is minimal among all sets such that (choose one):

(a) $\color{white} S$ is a compact connected set containing $\color{yellow} A$

(b) $\color{white} S$ is a compact connected set such that ${\color{white}S}\cup {\color{yellow}A}$ is also connected (Ducret-Troyanov, 2008)

(c) ${\color{white}S}\cup {\color{yellow}A}$ is connected (P.-Stepanov, 2012)

comparing the different formulations (1)

(a)     (b)     (c)

comparing the different formulations (2)

(a)  (b)  (c)

Existence

If $X$ is a proper complete connected metric space then minimizers exist.

Properties of minimizers, formulation (c)

If $\color{white} S$ is a minimizer with $\mathcal H^1({\color{white}S})<\infty$ then

  • $\color{white} S$ contains no loops
  • each connected component $C$ of ${\color{white}S}\setminus \color{yellow}A$ has $\mathcal H^1(\bar C) = \mathcal H^1(C) > 0$ and $\bar C$ has at most one single point on each connected component of $\color{yellow} A$
  • connected components of ${\color{white}S}\setminus {\color{yellow}A}$ are countable
  • ${\color{white}S}\setminus {\color{yellow}A}$ is a locally finite embedded tree composed by geodesic arcs

Example 1

$$ {{\color{yellow}A}}=\left\{\left(\frac 1 n,\frac 1 m\right)\colon n,m \in \mathbb N \cup \{\infty\}\right\} $$ Minimizers have infinite length.

Example 2



${\color{yellow}A} \subset [0,1]$ totally disconnected, uncountable, $\mathcal H^1({\color{yellow}A})>0$, ${\color{white}S}=[0,1]\setminus {\color{yellow}A}$ (Cantor-like) $$ \#\{\text{c.c. of } {\color{yellow}A}\} > \#\{\text{c.c. of } {\color{white}S}\} $$ $$ \mathcal H^1(\bar {\color{white}S}) > \mathcal H^1({\color{white}S}) $$

Example 3

If $\color{yellow} A$ is not bounded minimizers could not exist

Example 5 (open problem)

What if $\color{yellow} A$ is bounded but not closed?

In this case minimizers exist and ${\color{white}S}\cup {\color{yellow}A}$ is connected but not pathwise-connected.

The self-similar full binary tree Example

  • P-Stepanov-Teplitskaya (2015): $\color{white} S$ is the unique minimizer if $\lambda_n\to 0$ very fast.
  • Cherkashin-Teplitskaya (2023): $\color{white} S$ is a minimizer if $\lambda_n = \lambda^n$ with $\lambda$ sufficiently small.
Note: if $\lambda_n=\lambda^n$ the set $\color{yellow} A$ has a self-similarity dimension $d = \frac{\log 2}{-\log \lambda}$ which is $d>1$ for $\lambda>\frac 1 2$.

Proof (1)

Proof (2)

\[ \varepsilon_0 \le \varepsilon_1^+ + \varepsilon_1^- \le 2 \max\{\varepsilon_1^+,\varepsilon_1^-\} = 2 \varepsilon_1\\ \varepsilon_n \le \lambda_{n+1} \le \lambda^{n+1} \]

Proof (3)

Proof (4)

thank you!