In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.
The Reverse-log-Brunn-Minkowski inequality
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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 B\"or\"oczky-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.
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The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem
In the plane, for 0<q<2, equal dual curvature measures force origin-symmetric convex bodies to coincide; for q>n in R^n, distinct origin-symmetric bodies can share the same dual curvature measure.