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The Minkowski problem for the non-compact convex set with an asymptotic boundary condition
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abstract
In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for $\sigma$-finite measure on the conic domain $\Omega_C$.
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Cited by 2 Pith papers
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The Gaussian Minkowski-type problems for $C$-pseudo-cones
New existence and conditional uniqueness theorems for Gaussian Minkowski and log-Minkowski problems on C-pseudo-cones.
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The Gaussian-Minkowski problem for $C$-pseudo-cones
For every finite measure on the polar directions of a pointed cone, there exists a C-pseudo-cone whose Gaussian surface area measure equals that measure, with Gaussian co-volume no more than half the cone's.
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