A new differentiation, shape of the unit ball and perimeter measure

TL;DR

This paper introduces a novel blow-up method for computing perimeter measures in noncommutative nilpotent groups, revealing a surprising link between area formulas and the shape of the unit ball.

Contribution

It develops the first general formula for perimeter measure with respect to spherical Hausdorff measure in noncommutative nilpotent groups, connecting area formulas to the unit ball profile.

Findings

Established a new blow-up method for perimeter measure calculation.

Discovered a relationship between area formulas and unit ball shape.

Provided the first general perimeter measure formula in this context.

Abstract

We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. This result leads us to an unexpected relationship between the area formula with respect to a distance and the profile of its corresponding unit ball.