On tameness of zonoids (Q2066279)

A zonoid is a convex body that can be approximated in the Hausdorff metric by finite sums of line segments. Here are the precise definitions. A convex body is a nonempty, compact and convex subset of euclidean \(n\)-space. The support function of a convex body \(K\) in \(\mathbb{R}^n\) is the function \(h_k:S^{n-1}\to\mathbb{R}\) defined by \[ h_K(u):=\sup\{\langle u, x\rangle \mid x\in K\}. \] The convex body is a zonoid if there exists an even non-negative measure \(\mu\) on \(S^{n-1}\) such that \[ h_K(u)\frac{1}{2}\int_{S^{n-1}}|\langle u,x\rangle|d\mu(x). \] In dimension \(2\) every centrally symmetric convex body is a zonoid. In higher dimension no simple characterization exists. In the paper under review the authors work in the tame setting of o-minimality to show the following tame content of the concept of zonoids. The o-minimal structure \(\mathbb{R}_\mathrm{an}\) consisting of globally subanalytic sets and functions and its o-minimal expansion \(\mathbb{R}_{\mathrm{an},\exp}\) by the global exponential function are considered. Theorem. If a family of convex bodies is definable in \(\mathbb{R}_\mathrm{an}\), then the set of zonoids in this family is definable in \(\mathbb{R}_{\mathrm{an},\exp}\). The proof use the fact that a parameterized integral with integrand definable in \(\mathbb{R}_\mathrm{an}\) is definable in \(\mathbb{R}_{\mathrm{an},\exp}\) (see [\textit{J.-M. Lion} and \textit{J.-P. Rolin}, Ann. Inst. Fourier 48, No. 3, 755--767 (1998; Zbl 0912.32007); \textit{G. Comte} et al., Ill. J. Math. 44, No. 4, 884--888 (2000; Zbl 0982.32009); \textit{R. Cluckers} and \textit{D. J. Miller}, Duke Math. J. 156, No. 2, 311--348 (2011; Zbl 1216.26008)]. To apply this result analysis as the cosine transform is needed.