The Reverse-log-Brunn-Minkowski inequality
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916479600951296 |
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| author | Xi, Dongmeng |
| author_facet | Xi, Dongmeng |
| contents | Firstly, we propose our conjectured Reverse-log-Brunn-Minkowski inequality (RLBM).
Secondly, we show that the (RLBM) conjecture is equivalent to the log-Brunn-Minkowski (LBM) conjecture proposed by Böröczky-Lutwak-Yang-Zhang. We name this as ``reverse-to-forward" principle. Using this principle, we give a very simple new proof of the log-Brunn-Minkowski inequality in dimension two.
Finally, we establish the ``reverse-to-forward" principle for the log-Minkowski inequality (LM). Using this principle, we prove the log-Minkowski inequality in the case that one convex body is a zonoid (the inequality part was first proved by van Handle). Via a study of the lemma of relations, the full equality conditions (``dilated direct summands") are also characterized, which turns to be new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04266 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Reverse-log-Brunn-Minkowski inequality Xi, Dongmeng Metric Geometry 52A40 Firstly, we propose our conjectured Reverse-log-Brunn-Minkowski inequality (RLBM). Secondly, we show that the (RLBM) conjecture is equivalent to the log-Brunn-Minkowski (LBM) conjecture proposed by Böröczky-Lutwak-Yang-Zhang. We name this as ``reverse-to-forward" principle. Using this principle, we give a very simple new proof of the log-Brunn-Minkowski inequality in dimension two. Finally, we establish the ``reverse-to-forward" principle for the log-Minkowski inequality (LM). Using this principle, we prove the log-Minkowski inequality in the case that one convex body is a zonoid (the inequality part was first proved by van Handle). Via a study of the lemma of relations, the full equality conditions (``dilated direct summands") are also characterized, which turns to be new. |
| title | The Reverse-log-Brunn-Minkowski inequality |
| topic | Metric Geometry 52A40 |
| url | https://arxiv.org/abs/2307.04266 |