{
  "id": "2206.02123",
  "version": "v1",
  "published": "2022-06-05T08:38:39.000Z",
  "updated": "2022-06-05T08:38:39.000Z",
  "title": "On the volume of the Minkowski sum of zonoids",
  "authors": [
    "Matthieu Fradelizi",
    "Mokshay Madiman",
    "Mathieu Meyer",
    "Artem Zvavitch"
  ],
  "categories": [
    "math.MG",
    "math.FA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-05T08:38:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A40",
      "52A39",
      "52A20"
    ],
    "keywords": [
      "minkowski sum",
      "local alexandrov-fenchel inequality",
      "local loomis-whitney inequality",
      "brunn-minkowski theory",
      "convex geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}