Two-homogeneous measure projection

Let $d\ge 2$, let ${\mathbb{S}}^{d-1}=\{\theta \in {\mathbb{R}}^d:\mathrm{\mid }\theta \mathrm{\mid }=1\}$, and let $\mu$ be a finite nonnegative Borel measure on ${\mathbb{R}}^d$ with $\int \mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu (u)<\mathrm{\infty }$. Define $\rho (u)=u\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ for $u\mathrm{\ne }0$. The two-homogeneous projection of $\mu$ is the set function on Borel $A\subseteq {\mathbb{S}}^{d-1}$

$h^2(\mu )(A)={\int }_{{\mathbb{R}}^d\setminus \{0\}}{\mathbf{1}}_A(\rho (u))\mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu (u)\mathrm{.}$

It is a finite nonnegative Borel measure: the integrand defines the weighted measure $\mathrm{\mid }u{\mathrm{\mid }}^2\mu$ on ${\mathbb{R}}^d\setminus \{0\}$, and preimages under the continuous map $\rho$ preserve disjoint unions; its total mass is at most $\int \mathrm{\mid }u{\mathrm{\mid }}^{2}d\mu$. Mass at $u=0$ contributes zero. If $\mathrm{\Phi }$ or $V$ is positively two-homogeneous, this projection records its radial integral on the sphere; the exact identities are stated separately.

Feature-measure energy and its regularity assumptions

Chizat and Bach, Appendix A.2.2 and Appendix C.2.1

Ambrosio, Gigli, and Savaré, Chapter 8