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The Orlicz Aleksandrov problem for Orlicz integral curvature. (English) Zbl 1514.52006

Let \(K\) be a compact convex body that contains the origin in its interior. For \(\sigma \subset \partial K\), let \(\nu_K(\sigma) \subset S^{n-1}\) be the spherical image of \(\sigma\), consisting of all outer normal vectors with foot point in \(\sigma\). Let \(r_K:S^{n-1} \to \partial K\) be the radial projection. Alexandrov’s integral curvature is the measure on \(S^{n-1}\) defined by \[ J(K,\omega):=\mathcal H^{n-1}(\alpha_K(\omega)), \] where \(\alpha_K(\omega)=\nu_K(r_K(\omega))\).
The Alexandrov problem is: what are necessary and sufficient conditions for a measure \(\mu\) on \(S^{n-1}\) such that there exists a convex body containing the origin in its interior such that \(J(K,\bullet)=\mu\)? If \(K\) exists, is it unique? The problem was solved in [A. D. Aleksandrov, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 35, 131–134 (1942; Zbl 0061.37604)].
An \(L_p\)-version of this problem was considered in [Y. Huang et al., J. Differ. Geom. 110, No. 1, 1–29 (2018; Zbl 1404.35139)]. First, Alexandrov’s integral curvature is expressed in terms of a variation of the entropy functional. By replacing the entropy by an \(L_p\)-version, and the Minkowski sum by the \(L_p\)-sum, they define the \(L_p\)-integral curvature \(J_p(K,\bullet)\). They can solve the corresponding \(L_p\)-Alexandrov problem \(J_p(K,\bullet)=\mu\) if \(p\) is positive and if \(p\) is negative but \(\mu\) is an even measure.
In the paper under review, the authors go one step further and study an Orlicz version of Alexandrov’s problem. They first define a measure \(J_\phi(K,\bullet)\) that depends on the choice of a continuous function \(\phi:(0,\infty) \to (0,\infty)\) and that reduces to \(J_p(K,\bullet)\) in the case where \(\phi(t)=t^p\). Again this measure can be written in a variational way using a suitable notion of entropy. The existence part of the Orlicz-Alexandrov problem \(J_\phi(K\bullet)=\mu\) is then solved under the assumption that \(\mu\) is even and \(\phi\) satisfies one of the two following conditions:
1.
\(\phi(t)=\frac{-1}{t \varphi'(t)}\), where \(\varphi:(0,\infty) \to (0,\infty)\) is a continuous and strictly decreasing function that satisfies \(\lim_{t \to 0} \varphi(t)=\infty\), \(\lim_{t \to \infty} \varphi(t)=0\).
2.
\(\phi(t)=\frac{1}{t \varphi'(t)}\), where \(\varphi:(0,\infty) \to (0,\infty)\) is a continuous and strictly increasing function that satisfies \(\lim_{t \to 0} \varphi(t)=0\), \(\lim_{t \to \infty} \varphi(t)=\infty\).
Taking \(\varphi(t)=\frac{1}{p t^p}\), \(p>0\), in the first case and \(\varphi(t)=\frac{-1}{p t^p}\), \(p<0\), in the second case, this reduces to \(\phi(t)=t^p\), i.e. the \(L_p\)-Alexandrov problem.
The necessary and sufficient condition on \(\mu\) such that \(J_\phi(K,\bullet)=\mu\) can be solved is that \(\mu\) is not concentrated on any great subsphere of \(S^{n-1}\) in the first case and that \(\mu\) vanishes on all great subspheres in the second case.

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
52A39 Mixed volumes and related topics in convex geometry
53C65 Integral geometry
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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