Expectiles provide a smooth and naturally tail-sensitive alternative to quantiles, and have recently emerged as powerful tools for describing dispersion and asymmetry. In this talk, we develop a framework in which expectiles serve as the basis for measuring inequality, leading to a new class of expectile-based inequality indices that offer a natural geometric counterpart to classical Lorenz-Gini methodology.

The key observation is that comparisons in convex stochastic order can be expressed in terms of inclusions between suitably defined expectile regions. This allows distributional spread to be described geometrically through a nested family of regions capturing tail behavior. Building on this idea, we introduce law-invariant functionals obtained by integrating expectiles or inter-expectile ranges across asymmetry levels. These constructions give rise not only to generalized deviation and inequality measures, but also to expectile-based risk measures, while remaining fully consistent with convex-order comparisons and preserving a clear probabilistic interpretation.

A central aspect of the approach is a geometric representation of expectiles via a star-shaped set in the plane, whose boundary is traced by scaled expectiles. The area of this set naturally defines an inequality index, playing a role analogous to that of the Gini index, but arising from expectile geometry rather than from quantile-based Lorenz curves.

We also extend the construction to multivariate settings by defining expectile regions through directional projections, thereby obtaining inequality measures capable of capturing genuinely multidimensional heterogeneity. Finally, we discuss empirical implementation and computational aspects, showing that the proposed functionals can be evaluated efficiently in practice.