The Convex Peano Curve Does Exist
Adam Paszkiewicz

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics · Point processes and geometric inequalities
