The even Orlicz Minkowski problem (Q984880)

For a convex body \(K\), let \(h_K\) and \(S_K\) denote its support function and its surface area measure, respectively. Let \(\varphi:(0, \infty)\rightarrow (0, \infty)\) be a fixed continuous function. The aim of the paper is the study of the even \(L_{\varphi}\) Minkowski problem. Let \(\mu\) be an even finite Borel measure on the sphere which is not concentrated on a great subsphere. Does there exist an origin symmetric convex body \(K\) in \(\mathbb R^n\) such that \[ c\psi(h_K)dS_K=d\mu\; \] for some positive number \(c\)? The main results contained in the paper prove the existence for the even Orlicz Minkowski problem. As particular cases of these results, a solution to the even Minkowski problem is provided, as well as the solution for the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\), \(0<p\neq n\), and the solution to the even volume-normalized \(L_p\) Minkowski problem for all positive \(p\). The paper presents a new approach to both, the classical Minkowski problem and the even \(L_p\) Minkowski problem for \(p>0\).