General \((p,q)\)-mixed projection bodies
For \(K\in{\mathcal K}^n_o\) (the set of convex bodies in \({\mathbb R}^n\) containing the origin in the interior) and \(Q\in{\mathcal S}^n_o\) (the set of star bodies in \({\mathbb R}^n\)), and for \(p,q\in{\mathbb R}\), \textit{E. Lutwak} et al. [Adv. Math. 329, 85--132 (2018; Zbl 1388.52003)] have defined the \(L_p\) dual curvature measures \(\tilde C_{p,q}(K,Q,\cdot)\) by \[ \int_{S^{n-1}} g(v)\,\mathrm{d}\tilde C_{p,q}(K,Q,v) = \frac{1}{n} \int_{S^{n-1}} g(\alpha_K(u))h_K(\alpha_K(u))^{-p}\rho_K(u)^q\rho_Q(u)^{n-q}\,\mathrm{d}u\] for all continuous real functions \(g\) on the unit sphere \(S^{n-1}\), where \(\alpha,h,\rho\) are, respectively, the radial Gauss map, the support function, and the radial function. For \(p\ge 1\), \(q\in{\mathbb R}\), \(\tau\in[-1,1]\), the present authors define the general \((p,q)\)-mixed projection body \(\Pi_{p,q}^\tau(K,Q)\) by specifying its support function, \[ h\left (\Pi_{p,q}^\tau(K,Q),\cdot\right)^p = nc_{n,p}(\tau) \int_{S^{n-1}} \varphi_\tau(\langle \cdot,v\rangle)^p\,\mathrm{d}\tilde C_{p,q}(K,Q,v),\] where \(c_{n,p}(\tau)\) is an explicit constant, \(\varphi_\tau(t)=|t|+\tau t\), and \(\langle\cdot\,,\cdot\rangle\) is the scalar product. Special cases are the classical projection body and the \(L_p\) projection body. The paper finds out for which values of \(\tau\) the volume of \(\Pi_{p,q}^\tau(K,Q)\) or of its polar body attains its extreme values. It thus extends earlier results of \textit{C. Haberl} and \textit{F. E. Schuster} [J. Differ. Geom. 83, No. 1, 1--26 (2009; Zbl 1185.52005)].
scientific article; zbMATH DE number 7435407
scientific article