6533b7d8fe1ef96bd1269973

RESEARCH PRODUCT

Isometric embeddings of snowflakes into finite-dimensional Banach spaces

Enrico Le DonneErik WalsbergTapio Rajala

subject

30L05 46B85 54C25 54E40 28A80Pure mathematicsmetric spacesGeneral MathematicsMathematicsofComputing_GENERALBanach space01 natural sciencesfunctional analysisCardinalityMathematics - Metric GeometryDimension (vector space)0103 physical sciencesFOS: MathematicsMathematics (all)Mathematics::Metric Geometry0101 mathematicsSnowflakeNormed vector spaceMathematicsConcave functionApplied Mathematicsta111010102 general mathematicsnormiavaruudetMetric Geometry (math.MG)normed spacesmetriset avaruudetMetric spacefractalsfraktaalit010307 mathematical physicsfunktionaalianalyysiMathematics (all); Applied MathematicsVector space

description

We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.

yearjournalcountryeditionlanguage
2016-09-12Proceedings of the American Mathematical Society
https://doi.org/10.1090/proc/13778