The \(L_p\) Minkowski problem associated with the compatible functional \(\mathbf{F}\) (Q6581768)

Let \({\mathcal K}\) denote the space of nonempty, compact, convex sets in \({\mathbb R}^n\). The authors consider functionals (first introduced by \textit{A. Colesanti} and \textit{M. Fimiani} [Indiana Univ. Math. J. 59, No. 3, 1013--1040 (2010; Zbl 1217.31001)]) \({\mathbf F}:{\mathcal K}\to(0,\infty)\), which are positively homogeneous of some degree \(\alpha\not=0\) and such that there is a finite Borel measure \(\mu_{\mathbf F}(K,\cdot)\) on the unit sphere \({\mathbb S}^{n-1}\) satisfying \N\[\N{\mathbf F}(K) =\frac{1}{\alpha} \int_{{\mathbb S}^{n-1}} h_K\,\mathrm{d}\mu_{\mathbf F}(K,\cdot),\N\]\N\[\N\lim_{\varepsilon\to 0^+} \frac{{\mathbf F}(K+\varepsilon L)-{\mathbf F}(K)}{\varepsilon} = \int_{{\mathbb S}^{n-1}} h_L\,\mathrm{d}\mu_{\mathbf F}(K,\cdot)\N\]\Nfor \(K,L\in{\mathcal K}\), where \(h_K\) denotes the support function of \(K\). The functional \({\mathbf F}\) is called compatible if (i) \(\alpha>0\), (ii) \({\mathbf F}\) is translation invariant, (iii) \({\mathbf F}\) is monotone under set inclusion, (iv) \({\mathbf F}\) satisfies a Brunn-Minkowski-type inequality with exponent \(1/\alpha\), with equality characterizing homothety, (v) \({\mathbf F}(K)=0\) if the volume of \(K\) is zero, (vi) \(\mu_{\mathbf F}(K,\cdot)\) is weakly continuous in \(K\). In spite of these numerous restrictions, there are important examples: the volume (\(\alpha=n\)), the \({\mathfrak p}\)-capacity for \(1\le {\mathfrak p} <n\) (\(\alpha= n-{\mathfrak p}\)), the torsional rigidity (\(\alpha=n+2\)). \N\NThe paper deals with Minkowski type problems for the measure \(\mu_{{\mathbf F},p}(K,\cdot)\) defined by \(\mathrm{d}\mu_{{\mathbf F},p}(K,\cdot)=h_K^{1-p}\mathrm{d}\mu_{\mathbf F}(K,\cdot)\). Let \({\mathbf F}\) and \(\mu_{\mathbf F}\) satisfy the assumptions listed above, let \(\mu\) be a finite Borel measure on \({\mathbb S}^{n-1}\) with support not concentrated on a closed hemisphere. The main results are: (1) If \(0<p<1\), then there exists a convex body \(K\) containing the origin such that \(\mu=\mu_{{\mathbf F},p}(K,\cdot)\). (2) If \(p\ge 1\), then there exists a convex body \(K\) containing the origin such that \(\mu={\mathbf F}(K)^{-1}\mu_{{\mathbf F},p}(K,\cdot)\). The proofs first solve the discrete case and then use approximation.