逆向-log-Brunn-Minkowski不等式
The Reverse-log-Brunn-Minkowski inequality
AI总结：
本文提出了逆向-log-Brunn-Minkowski不等式猜想，证明了其与正向LBM猜想的等价性并建立“逆向到正向”原理，借此给出了二维LBM不等式及zonoid情形下LM不等式的新证明与等号条件刻画。
AI中文摘要：
首先，我们提出猜想中的逆向-log-Brunn-Minkowski不等式（RLBM）。其次，我们证明（RLBM）猜想等价于Böröczky-Lutwak-Yang-Zhang提出的log-Brunn-Minkowski（LBM）猜想。我们将其命名为“逆向到正向”原理。利用该原理，我们给出了二维情形下log-Brunn-Minkowski不等式的一个非常简单的新证明。最后，我们为log-Minkowski不等式（LM）建立了“逆向到正向”原理。利用该原理，我们证明了一个凸体为zonoid情形下的log-Minkowski不等式（不等式部分最初由van Handle证明）。通过对关系引理的研究，完整的等号成立条件（“dilated direct summands”）也得到了刻画，这是全新的结果。
英文摘要：
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.
