Minkowski valuations on \(L^p\)-spaces (Q2844835)

Let \( \mathcal{K}^n \) be the collection of all convex bodies in \( {\mathbb R}^n \), and \( L^p ( {\mathbb R}^n , |x| \text{d} x) \) be the space of measurable functions \( f \) with finite norm \( \bigl(\int\nolimits_{{\mathbb R}^n} |f (x)|^p |x| \text{d} x \bigr)^{1/p} \). For \( f \in L^1 ( {\mathbb R}^n , |x| \text{d} x)\), the moment vector \( \overline{m} (f)\) and the moment body \( \overline{ M } (f)\) are defined as \( \overline{m} (f) = \int\nolimits_{{\mathbb R}^n} f(x) \text{d} x\), and, respectively, as the convex body with support function \( h(\overline{M} (f), u) = \int\nolimits_{{\mathbb R}^n} |f(x) | | x \cdot u| \text{d} x\) for \( u \in {\mathbb R}^n\).NEWLINENEWLINEA function \( \Phi: L^p ( {\mathbb R}^n , |x| \text{d} x) \to \mathcal{K}^n \) is called a Minkowski valuation if \( \Phi(f\wedge g) + \Phi (f \vee g) = \Phi (f) + \Phi (g) \), where \(f\wedge g = \min \{ f, g\} \), \(f\vee g = \max \{ f, g\} \), \(+\) denotes Minkowski addition, and \( \Phi (0) = \{ o\}\). The main results of the paper areNEWLINENEWLINETheorem 1.1. A function \( \Phi \), where \( n \geq 2\), is a continuous \(\text{GL}(n)\)-invariant Minkowski evaluation if and only if there exist continuous functions \( G: {\mathbb R}^n \to {\mathbb R}^n\) and \( H: {\mathbb R}^n \to {\mathbb R}^n\) such that \( |G(\alpha)| \leq \gamma |\alpha |^p \) and \( 0 \leq H(\alpha) \leq \delta |\alpha |^p \) for some non-negative real \( \gamma\), \(\delta \) and all \( \alpha \in {\mathbb R}^n\), and \( \Phi (f) = \overline{m} (G \circ f) + \overline{M} (H \circ f) \) for every \( f \in L^p ( {\mathbb R}^n , |x| \text{d} x)\).NEWLINENEWLINETheorem 1.2. A function \( \Phi\), where \( n \geq 2\), is a continuous \(\text{GL}(n)\)-invariant homogeneous Minkowski evaluation if and only if there exist real constants \( a, b \) with \( b \geq 0 \) such that \( \Phi (f) = a \overline{m} (| f|^p ) + b \overline{M} (|f|^p) \) for all \( f \in L^p ( {\mathbb R}^n , |x| \text{d} x)\).