The Convex Peano Curve Does Exist

TL;DR

This paper extends Peano's original space-filling curve by constructing a convexity-preserving continuous surjection from the interval to any convex closed set in R^2, demonstrating a new class of space-filling curves.

Contribution

It introduces a novel convexity-preserving space-filling curve that generalizes Peano's original construction to arbitrary convex sets.

Findings

Existence of a continuous surjection from [0,1] to any convex set in R^2.

The image of any subinterval is a convex set.

Generalization of Peano's curve with convexity properties.

Abstract

We refer here to the surprising construction made by Giuseppe Peano in 1890. He gave an example of a continuous function (called now the Peano curve) from the unit interval to the whole unit square. We show here the existence of a more general space-filling curve with additional convexity properties. More precisely: by T ? R2 we denote a convex closed and bounded set and we show that there exists a continuous surjection f : [0; 1] 7! T for which the image of any interval is a convex set.