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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)].

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