A definition of fractional k-dimensional measure: bridging the gap between fractional length and fractional area

TL;DR

This paper introduces a fractional $k$-dimensional measure that interpolates between fractional length and fractional area, converging to the classical Hausdorff measure as the fractional parameter approaches 1.

Contribution

It defines a new fractional measure for $k$-dimensional sets that unifies existing fractional notions of length and area, with a precise convergence result.

Findings

The fractional measure depends on a parameter $\sigma$ between 0 and 1.

Multiplying by $(1-\sigma)$, the measure converges to the Hausdorff measure as $\sigma$ approaches 1.

The measure coincides with classical notions for specific values of $k$.

Abstract

Here we introduce a fractional notion of $k$-dimensional measure, $0\leq k<n$, that depends on a parameter $\sigma$ that lies between $0$ and $1$. When $k=n-1$ this coincides with the fractional notions of area and perimeter, and when $k=1$ this coincides with the fractional notion of length. It is shown that, when multiplied by the factor $1-\sigma$, this $\sigma$-measure converges to the $k$-dimensional Hausdorff measure up to a multiplicative constant that is computed exactly. We also mention several future directions of research that could be pursued using the fractional measure introduced.