The generalized Gaussian Minkowski problem (Q6604719)

Let \(K\) be a convex body, i.e. a compact convex set with non-empty interior. The surface area measure \(S_K\) of \(K\) is defined as \N\[\NS_K(\omega)= \mathcal H^{n-1} (\nu_K^{-1}(\omega))\N\]\Nfor \(\omega\) a Borel set of \(S^{n-1}\), where \(\nu_K\) is the Gauss map of \(K\) and \(\mathcal H^{n-1}\) the Hausdorff measure. The Minkowski problem asks for a characterization of all measures on \(S^{n-1}\) occurring as surface area measures. The surface area measure satisfies \N\[\N\lim_{\epsilon \to 0} \frac{V_n(K+ \epsilon L)- V_n(K)}{\epsilon} = \int_{S^{n-1}} h_L(u) dS_K(u)\N\]\Nfor convex bodies \(L\) with support function \(h_L\).\N\NIn this paper the authors investigate the generalized Gaussian Minkowski problem. A generalized Gaussian measure \(G_{\alpha, q}\) on \(\mathbb R^n\) is given by its density \((1-\frac q{\alpha} |x|^\alpha)_+^{\frac 1q - \frac n{\alpha}-1}\) for \(q \neq 0\), and \(e^{\frac 1\alpha |x|^\alpha}\) for \(q=0\), with a suitable normalization. Replacing volume \(V_n\) by generalized Gaussian measure defines a generalized Gaussian surface area measure via \N\[\N\lim_{\epsilon \to 0} \frac{G_{\alpha, q}(K+ \epsilon L)- G_{\alpha, q}(K)}{\epsilon} = \int_{S^{n-1}} h_L(u) dS_{\alpha,q,K}(u),\N\]\Nand the generalized Gaussian Minkowski problem asks for a characterization of all measures on \(S^{n-1}\) occurring as generalized Gaussian surface area measures.\N\NThe authors prove that for certain ranges of \(q\) and \( \alpha\), given a measure on \(S^{n-1}\) there is a convex body solving the generalized Gaussian Minkowski problem up to normalization.\N\NMoreover, replacing Minkowski addition by the \(L_p\)-Minkowski addition leads to the \(L_p\)-surface area measure \(S_{p,K}\) and the \(L_p\)-Minkowski problem, and the authors also investigate the generalized Gaussian \(L_p\)-Minkowski problem.