On P. McMullen's Conjecture on Translation Invariant Valuations

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Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations (2002, background): “Thus it is equal to f(E) ·volE , where volE is the volume form on E defined by the metric on V , and f(E) is a constant depending on E.”
Range characterization of the cosine transform on higher Grassmannians (2001, methodology): “This step uses connection of the cosine transform with the theory of valuations on convex sets (discussed in Section 1) and earlier results of one of the authors about representation theoretical properties of valuations [1], [2].”
Valuations on Polytopes Containing the Origin in Their Interiors (2002, background): “Further contributions to this problem include results by Schneider [22], Klain [13], and Alesker [ 5 ].”
A Hadwiger-Type Theorem for the Special Unitary Group (2008, background): “The theory of translation invariant continuous valuations is a very rich one, see for instance [22], [1], [2], [20], [13], [8].”
The space of isometry covariant tensor valuations (2007, background): “If we put Ip,k := {(l, m, r, s) : l, r, s ∈ N0, m ∈ {0, . . . , n − 1} or (m, s) = (n, 0), 2l + r + s = p, m + r = k}, then Bp,k := {QlΦm,r,s : (l, m, r, s) ∈ Ip,k} is the set of basic tensor valuations contained in Tp,k. Alesker’s characterization theorem can be stated as Tp,k = linBp,k.”