Minkowski sums of three dimensional projections of convex bodies (Q2707495)

Let \({\mathcal K}^d\) denote the space of convex bodies in \({\mathbb R}^d\). For \(K\in{\mathcal K}^d\), let \(P_k(K)\) denote the average (in the sense of Minkowski sums) of the projections of \(K\) onto \(k\)-dimensional subspaces of \({\mathbb R}^d\). Here, \(k\in\{1,\ldots,d-1\}\) and the average is formed by integration of support functions with respect to the rotation invariant probability measure on the corresponding Grassmannian manifold. NEWLINENEWLINENEWLINEThe subject of the paper are injectivity properties of the operator \(P_k\), that is the question whether a convex body is determined by the average of its \(k\)-dimensional projections. A positive answer for \(k\geq d/2\) has recently been given by \textit{P. Goodey} [Mathematika 45, No. 2, 253-268 (1998; Zbl 0959.52003)], where also the difficult case \(k=2\) has been treated. The operator \(P_2\) turns out to be injective in all dimensions except \(14\). NEWLINENEWLINENEWLINEThe main result of the present contribution states that \(P_k\) is injective also for \(k=3\). For the proof, the authors show that the multipliers of an associated (intertwining) linear operator on \(L^2(S^{d-1})\) are never zero, which requires delicated vips calculations involving Gegenbauer polynomials. The paper concludes with some useful observations about the range of \(P_k\) for general \(k\).NEWLINENEWLINEFor the entire collection see [Zbl 0948.00038].