Products of snowflaked euclidean lines are not minimal for looking down

TL;DR

This paper demonstrates that certain snowflaked Euclidean line products are not minimal for looking down, using a novel distance construction and extending methods from geometric analysis to fractal and metric space theory.

Contribution

It introduces a new distance method showing that products of snowflaked Euclidean lines are not minimal for looking down, extending prior work on the Heisenberg group.

Findings

Products of snowflaked Euclidean lines are not minimal for looking down.

A new distance is constructed to demonstrate the non-minimality.

The approach extends techniques from analysis of the Heisenberg group.

Abstract

We show that products of snowflaked Euclidean lines are not minimal for looking down. This question was raised in Fractured fractals and broken dreams, Problem 11.17, by David and Semmes. The proof uses arguments developed by Le Donne, Li and Rajala to prove that the Heisenberg group is not minimal for looking down. By a method of shortcuts, we define a new distance $d$ such that the product of snowflaked Euclidean lines looks down on $(\mathbb R^N,d)$, but not vice versa.