{
  "id": "2206.10494",
  "version": "v1",
  "published": "2022-06-21T16:06:43.000Z",
  "updated": "2022-06-21T16:06:43.000Z",
  "title": "Characterizing Distances Between Points in the Level Sets of a Class of Continuous Functions on a Closed Interval",
  "authors": [
    "Yuanming Luo",
    "Henry Riely"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Given a continuous function $f:[a,b]\\to\\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide $[a,b]$. Additionally, we show that it must contain at least one third of the interval $[0,b-a]$. Lastly, we explore some higher dimensional generalizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-21T16:06:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A06"
    ],
    "keywords": [
      "continuous function",
      "level sets",
      "characterizing distances",
      "closed interval",
      "higher dimensional generalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}