Projection functions, area measures and the Alesker–Fourier transform

Abstract

Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are constructed.

Fractional-order projection of a chaotic system with hidden attractors and its passivity-based synchronization

“…For instance, Sheu and Capoferri et al demonstrated quantum projections and pseudo-differential projections, respectively [19,20]. Lau et al and Dorrek et al revealed projections in Banach algebras and projection functions of the Alesker-Fourier transform, respectively [21,22]. Next, Basso reported both maximal projection constants and maximal projections in a threedimensional subspace [23].…”

Lutwak-Petty projection inequalities for Minkowski valuations and their duals

“…One of its central inequalities is the Busemann intersection inequality [7], which relates the volume of a star body to that of its intersection body. Intersection bodies were first introduced by Lutwak in [37] and ever since a number of authors has contributed to the research on the duality between projection and intersection bodies (confer [15,18,57] for more details). Recently it was shown by Lu and Leng [27] that inequalities analogous to the Busemann intersection inequality also hold for intersection bodies of all orders.…”

“…15), we have for K ∈ K n o and L ∈ S n o ,V p (K, Γ µ p L) = 1 n S n−1 h(Γ µ p L, u) p dS p (K, u) = 1 n(n + p)V n (L) S n−1 S n−1 h(Z µ p (u), v) p ρ(L, v) n+p dv dS p (K, u).Using Fubini's theorem, definition (3.12) of Φ µ p , and (2.13) yields(n + p)V n (L) 2 V p (K, Γ µ p L) = 1 n S n−1 h(Φ µ p K, v) p ρ(L, v) n+p dv = Ṽ−p (L, Φ µ, * p K). (5.7)Taking now K = Γ µ p L, we obtainV n (Γ µ p L) = 2 (n + p)V n (L)Ṽ−p (L, Φ µ, * p Γ µ p L).…”