{
  "id": "2312.16923",
  "version": "v1",
  "published": "2023-12-28T09:42:57.000Z",
  "updated": "2023-12-28T09:42:57.000Z",
  "title": "Radon transforms with small derivatives and distance inequalities for convex bodies",
  "authors": [
    "Julián Haddad",
    "Alexander Koldobsky"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\\in \\mathbb N$, any $q\\in [0,n-1)$ which is not an odd integer, any origin-symmetric convex body $K$ of volume one in $\\mathbb R^n$ and any infinitely smooth probability density $f$ on $K$ we have $$\\max_{\\xi \\in S^{n-1}} {\\frac 1{\\cos(\\pi q/2)}\\mathcal R f(\\xi, \\cdot)_t^{(q)}(0)} \\ge \\left( \\frac {c(q+1)}{n}\\right)^{\\frac{q+1}2}.$$ Here $\\mathcal R f(\\xi,t)$ is the Radon transform of $f$, and the fractional derivative of the order $q$ is taken with respect to the variable $t\\in \\mathbb R$ with fixed $\\xi\\in S^{n-1}.$ In this note we show that there exist an origin-symmetric convex body $K$ of volume 1 in $\\mathbb R^n$ and a continuous probability density $g$ on $K$ so that $$\\max_{\\xi\\in S^{n-1}} {\\frac 1{\\cos(\\pi q/2)}\\mathcal R g(\\xi, \\cdot)_t^{(q)}(0)} \\leq \\frac 1{\\sqrt n} (c(q+1))^{\\frac{q+1}2}.$$ In the case $q=0$ this was proved in [5,6], and it was used there to obtain a lower estimate for the maximal outer volume ratio distance from an arbitrary origin-symmetric convex body $K$ to the class of intersection bodies. We extend the latter result to the class $L_{-1-q}^n$ of bodies in $\\mathbb R^n$ that embed in $L_{-1-q}.$ Namely, for every $q\\in [0,n)$ there exists an origin-symmetric convex body $K$ in $\\mathbb R^n$ so that ${d_{\\operatorname{ovr}}}(K, L_{-1-q}^n) \\ge c n^{\\frac 1{2(q+1)}}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-28T09:42:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "44A12"
    ],
    "keywords": [
      "radon transform",
      "small derivatives",
      "distance inequalities",
      "inequality",
      "maximal outer volume ratio distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}