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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.12802 |
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Table of Contents:
- 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 $σ$-finite measure on the conic domain $Ω_C$.