On the volume of the Minkowski sum of zonoids
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929272296308736 |
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| author | Fradelizi, Matthieu Madiman, Mokshay Meyer, Mathieu Zvavitch, Artem |
| author_facet | Fradelizi, Matthieu Madiman, Mokshay Meyer, Mathieu Zvavitch, Artem |
| contents | We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establish an improved inequality in ${\mathbb R^2}$. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the $L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case $p=2$, in any dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_02123 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the volume of the Minkowski sum of zonoids Fradelizi, Matthieu Madiman, Mokshay Meyer, Mathieu Zvavitch, Artem Metric Geometry Functional Analysis 52A40, 52A39, 52A20 We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establish an improved inequality in ${\mathbb R^2}$. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the $L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case $p=2$, in any dimension. |
| title | On the volume of the Minkowski sum of zonoids |
| topic | Metric Geometry Functional Analysis 52A40, 52A39, 52A20 |
| url | https://arxiv.org/abs/2206.02123 |