On lower estimations of square-linear ratio for plane Peano curves

TL;DR

This paper proves lower bounds on the square-linear ratio for mappings of a segment into a square, showing that distances can be significantly distorted, especially under certain boundary conditions.

Contribution

It establishes new lower bounds for the square-linear ratio of plane Peano curves, including cases with boundary constraints.

Findings

The square of the Euclidean distance can exceed the segment distance by at least 3.625 times.

With boundary conditions, the ratio increases to at least 4 plus an arbitrarily small epsilon.

These bounds are proven for any mapping of a segment into a square.

Abstract

It is proved that for any mapping of a unit segment to a unit square, there is a pair of points of the segment for which the square of the Euclidean distance between their images exceeds the distance between them on the segment by at least $3\frac58$ times. And the additional condition that the images of the beginning and end of the segment belong to opposite sides of the square increases the estimate to $4+\varepsilon$.