{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1V11QM5C07"
    },
    {
      "field": "uses",
      "target": "AJ5RG9XQYZ"
    },
    {
      "field": "about",
      "target": "AJ5RG9XQYZ"
    },
    {
      "field": "defines",
      "target": "PXF8731MBK"
    }
  ],
  "body": "\n## Summary\n\nA normed space is uniformly convex when midpoints of two unit vectors that are far apart lie uniformly deep inside the unit ball.\nIt is a quantitative form of [[strict convexity|GDTKW9KPH8]] of the unit ball.\nHilbert spaces and $L^p$ spaces for $1 \u003c p \u003c \\infty$ are uniformly convex, and the [[Milman–Pettis theorem|AJ5RG9XQYZ]] says that every uniformly convex Banach space is reflexive.\n\n## Explanations\n\n### L1 human\n\nPicture the unit ball as a round ball.\nIf two points on or in the ball are a fixed distance apart, the midpoint of the segment between them is noticeably inside the ball, not near its edge.\nUniform convexity asks that this happen with a margin that depends only on the distance between the points.\nA square ball in the plane fails this: the midpoint of two corners on the same side lies on the boundary.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46B20"
    ],
    "name": "Uniformly convex space",
    "same_as": {
      "wikidata": "Q847535",
      "wikipedia": "Uniformly convex space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Uniformly_convex_space"
      }
    ],
    "specializes": [
      "GDTKW9KPH8"
    ],
    "tag": "BEAMC5011M"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/BEAMC5011M/",
  "links": [
    {
      "field": "specializes",
      "target": "GDTKW9KPH8"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/BEAMC5011M.md",
  "msc": [
    "46B20"
  ],
  "name": "Uniformly convex space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BEA/BEAMC5011M.md",
  "status": null,
  "tag": "BEAMC5011M",
  "uri": "m2://BEAMC5011M",
  "url": "https://math2.tamnd.com/wiki/uniformly-convex-space/",
  "writing": {
    "tag": "BEAMC5011M",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}