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

Abstract

Here we introduce a notion of fractional k-dimensional measure, \(0\le k<n\), that depends on a parameter \(\sigma \) that lies between 0 and 1. When \(k=n-1\) this coincides with the notions of fractional area and perimeter, and when \(k=1\) this coincides with the notion of fractional 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.