Tensor valuations on convex bodies and integral geometry (Q2704114)

\textbf{V.Oganyan (Erevan)}: Let \(E^n\) be \(n\)-dimensional vector space, \(x\in E^n\). For the moment vector \(z_{n+1}(K) = \int_K x dx\) of an \(n\)-dimensional convex body \(K\) and \(\lambda>0\), there is a Steiner formula \(z_{n+1}(K + \lambda B^n) = \sum_{i=0}^n \binom{n}{i} q_i(K) \lambda^i\) [see the author, Convex bodies: The Brunn-Minkowski theory (1993; Zbl 0798.52001) p. 304], which defines quermassvectors \(q_i\), \(i=0,\dots,n\). Characterization theorems and integral-geometric formulae corresponding to those for quermassintegrals were obtained in the mentioned book. The natural next step is to consider the general moment tensor \(M_r(K) = \int_K x\otimes\dots\otimes x dx\) for \(r\in N\) and to derive a Steiner formula for it. The author investigates the coefficients in the Steiner formula for general moment tensors. They are tensor-valued valuations. The topic of the last section is to study integral-geometric formulae of kinematic type for general moment tensors and functions derived from them. NEWLINENEWLINENEWLINE\textbf{Maria Moszyńska (Warszawa)}: The paper concerns valuations on the class of convex bodies in \({\mathbb{R}}^n\), with values of the valuations being tensors of an arbitrary order \(r \geq 0\).NEWLINENEWLINE The author defines the moment tensor \(M_r\) of order \(r\) and proves the Steiner type formula, by which the coefficient tensors \(M_{r,i}\) are determined (Section 3). This is a natural generalization of the well known notions: for \(r=0\) and \(r=1\) the tensors \(M_{r,i}\) are quermass integrals (i.e. intrinsic volumes) and quermassvectors, respectively.NEWLINENEWLINEIn Section 4 the author investigates the formulae of integral geometry for \(M_r\) and \(M_{r,i}\) (in particular, the kinematic formula). These results extend those obtained by \textit{H. R. Müller} [Rend. Circ. Mat. Palermo (2) 2, 1-21 (1953)] for \(n=r=2\).NEWLINENEWLINE Tensor valuations were studied also by \textit{P. McMullen} [Rend. Circ. Mat. Palermo Suppl. (2) 50, 259-271 (1997; Zbl 0901.52009)], \textit{K. Przesławski} [Rend. Circ. Mat. Palermo Suppl. (2) 50, 299-314 (1997; Zbl 0904.52003)], and recently by \textit{S. Alesker} [Ann. of Math. (2) 149, 977-1005 (1997; Zbl 0941.52002)]. The Schneider approach is quite different from the previous ones: he starts from sufficiently smooth convex bodies (instead of polytopes) and then extends his results to arbitrary convex bodies by approximation.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00040].