Covariogram and orientation-dependent chord lenght distribution function for oblique prism (Q2676598)

Let \(\mathbb{R}^{n}\) be the \(n\)-dimensional Euclidean space, \(D\subset \mathbb{R}^{n} \) a bounded convex body with interior points and \(L_{n} (\cdot )\) the \(n\)-dimensional Lebesgue measure in \(\mathbb{R}^{n}\). If \(x\in\mathbb{R}^n\), \(D + x\) denote the translate of \(D\) by \(x\), that is, \(D + x = \{y + x, y\in D\}\). If \(D\subset\mathbb{R}^n\) is a convex body, then its covariogram \(C(D, x)\) is the function defined for \(x \in\mathbb{R}^n\) by \(C(D, x) = L_n(D \cap (D + x))\). G. Matheron conjectured that covariogram of a convex body specify it in the class of all convex bodies, up to translations and reflections (reflection means reflection with respect to a point). This hypothesis is known as G. Matheron's hypothesis, [\textit{R. Schneider} and \textit{W. Weil}, Stochastic and integral geometry. Berlin: Springer (2008; Zbl 1175.60003)]. \textit{G. Averkov} and \textit{G. Bianchi} confirmed Matheron's conjecture for \(n=2\), see [J. Eur. Math. Soc. (JEMS) 11, No. 6, 1187--1202 (2009; Zbl 1185.52002)]. In the plane the positive answer for the Matheron's hypothesis in the class of convex polygons received by \textit{W. Nagel} [J. Appl. Probab. 30, No. 3, 730--736 (1993; Zbl 0781.60018)]. G. Bianchi has also proved that for \(n\ge4\) the hypothesis is false. It is known that centrally symmetric convex bodies in any dimension, are determined by their covariogram up to translations. For \(n=3\) the problem is open. Nevertheless, for the case of bounded convex polyhedron for \(n=3\) Matheron's conjecture received a positive answer. Note that convexity is essential for this set of problems. An example of two non-congruent and non-convex polygons with the same covariogram has been constructed, see [\textit{C. Benassi} et al., Mathematika 56, No. 2, 267--284 (2010; Zbl 1198.52007)]. Thus, investigation of the covariogram of three dimensional convex bodies becomes an important first step in the study of Matheron's conjecture in \(\mathbb{R}^{3}\). Note that the explicit form for the covariogram of three dimensional convex bodies has been known only in the case of a ball. In the present paper, the author obtains the following results: \begin{itemize} \item[(1)] The calculation of the covariogram and orientation-dependent chord length distribution function for any trapezoid. This is a generalization of the result of [\textit{V. K. Ohanyan} and \textit{D. M. Martirosyan}, J. Contemp. Math. Anal., Armen. Acad. Sci. 55, No. 6, 344--355 (2020; Zbl 1454.60023)]. \item[(2)] Relationships between the covariogram and the orientation-dependent chord length distribution function of an oblique prism and those of its base. \item[(3)] Explicit forms of the covariogram and the orientation-dependent chord length distribution function of an oblique prism with cyclic, elliptical, trapezoid and triangular bases. \end{itemize} The second and third results are a generalization of [\textit{H. S. Harutyunyan} and \textit{V. K. Ohanyan}, J. Contemp. Math. Anal., Armen. Acad. Sci. 49, No. 6, 366--375 (2014; Zbl 1354.60013); translation from Izv. Nats. Akad. Nauk Armen., Mat. 49, No. 6, 3--15 (2014)], see also [\textit{H. S. Harutyunyan} and \textit{V. K. Ohanyan}, J. Contemp. Math. Anal., Armen. Acad. Sci. 49, No. 3, 139--156 (2014; Zbl 1316.60022); translation from Izv. Nats. Akad. Nauk Armen., Mat. 49, No. 3, 3--24 (2014)].