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On the regularity of solutions to a generalization of the Minkowski problem. (English) Zbl 0867.52003
W. Firey and E. Lutwak generalized the classical Brunn-Minkowski theory of convex bodies to a \(p\)-theory \((p\geq 1)\) introducing a linear combination \(\lambda\cdot K +_p\mu\cdot L\) by the support function \(h(\lambda\cdot K +_p\mu\cdot L,\cdot):=(\lambda h(K,\cdot)^p+\mu h(L,\cdot)^p)^{\frac1p}\) for \(\lambda\geq 0\), \(\mu\geq 0\) (not both zero) and convex bodies \(K, L\in\mathbb{R}^n\) containing the origin in their interior. From this a \(p\)-mixed volume \(V_p(K,L)\) may be derived by its equality with \[ \frac pn \lim_{\varepsilon\to+0} \frac{V(K +_p\varepsilon\cdot L) - V(K)}{\varepsilon}=\frac1n \int_{S^{n-1}}h(L, u)^p h(K,u)^{1-p} dS(K,u) \] where \(S(K,\cdot)\) is Aleksandrov’s surface area measure of \(K\) on the sphere \(S^{n-1}\) such that \(f_p(K,u):= h(K,u)^{1-p} f(K,u)\) can be considered as \(p\)-curvature function of \(K\) if \(S(K,\cdot)\) is absolutely continuous with respect to the Lebesgue measure \(\omega\) on \(S^{n-1}: dS(K,u)= f(K,u)=d\omega(u)\). In a former paper E. Lutwak [J. Differ. Geom. 38, No. 1, 131-150 (1993; Zbl 0788.52007)] solved the “generalized Minkowski problem” \(h(K,u)^{1-p} f(K,u)=g(u)\) under the restrictive assumptions that \(1-n\neq 1- p\leq 0\) and that \(g\) is a prescribed even, positive and continuous function on \(S^{n-1}\) showing that there exists a unique centered convex body \(K\) with the \(p\)-curvature function \(g\).
In the present paper the authors show that if hereby additionally \(g\) is of class \(C^m(S^{n-1})\) \((m\geq 3)\) then the support function \(h=h(K,\cdot)\) of the solution \(K\) is of class \(C^{m+1,\alpha}(S^{n-1})\) for any \(\alpha\in (0,1)\). Furthermore, if \(g\) is analytic, then \(h\) is analytic as well. The proof uses a continuity scheme as it has been applied by L. Nirenberg [Commun. Pure Appl. Math. 6, 337-394 (1953; Zbl 0051.12402)].

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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