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- W4300697932 abstract "Log-Brunn-Minkowski inequality was conjectured by Boroczky, Lutwak, Yang and Zhang cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequality for log-concave measures, which in the case of Gaussian measure was conjectured by Gardner and Zvavitch cite{GZ}. In this note, we obtain stability results for both Log-Brunn-Minkowski and dimensional Brunn-Minkowski inequalities for rotation invariant log-conave measures near a ball. Remarkably, the assumption of symmetry is only necessary for Log-Brunn-Minkowski stability, which emphasizes an important difference between the two conjectured inequalities. Also, we determine the infinitesimal version of the log-Brunn-Minkowski inequality. As a consequence, we obtain a strong Poincar'{e}-type inequality in the case of unconditional convex sets, as well as for symmetric convex sets on the plane. Additionally, we derive an infinitesimal equivalent version of the B-conjecture for an arbitrary measure." @default.
- W4300697932 created "2022-10-04" @default.
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- W4300697932 date "2016-06-18" @default.
- W4300697932 modified "2023-10-14" @default.
- W4300697932 title "On the stability of Brunn-Minkowski type inequalities" @default.
- W4300697932 doi "https://doi.org/10.48550/arxiv.1606.06586" @default.
- W4300697932 hasPublicationYear "2016" @default.
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