Isometric Embeddability of Snowflakes
Erik Walsberg
TL;DR
This paper proves that snowflake metric spaces with positive Hausdorff dimension cannot be embedded into Euclidean space without distortion.
Contribution
It establishes a fundamental limitation on isometric embeddings of snowflake metrics into Euclidean spaces.
Findings
Snowflakes with positive Hausdorff dimension are not isometrically embeddable into Euclidean space.
The result clarifies the geometric constraints of snowflake metrics.
Provides a theoretical boundary for embedding snowflake spaces.
Abstract
We show that a snowflake of a metric space with positive Hausdorff dimension does not admit an isometric embedding into euclidean space.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Mathematics and Applications · Computational Geometry and Mesh Generation