{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[normed vector space|03J5RAKB30]] over $\\R$ or $\\C$.\nThe space $X$ is *strictly convex* if for all $x, y \\in X$ with $\\norm{x} = \\norm{y} = 1$ and $x \\ne y$,\n$$\\left\\| \\frac{x + y}{2} \\right\\| \u003c 1.$$\n\n## Remarks\n\nEquivalent forms: the unit sphere contains no nondegenerate line segment; and $\\norm{x + y} = \\norm{x} + \\norm{y}$ with $x, y \\ne 0$ implies $y = c x$ for some real $c \u003e 0$.\nExample: $\\R^2$ with the Euclidean norm.\nIf $\\norm{x} = \\norm{y} = 1$ and $x \\ne y$, the parallelogram law gives $\\norm{(x + y)/2}^2 = 1 - \\norm{(x - y)/2}^2 \u003c 1$.\nNon-example: $\\R^2$ with $\\norm{(a, b)}_1 = \\abs{a} + \\abs{b}$.\nFor $x = (1, 0)$ and $y = (0, 1)$ the midpoint $(1/2, 1/2)$ has norm $1$.\nEvery [[uniformly convex space|BEAMC5011M]] is strictly convex, but not conversely.\nFor instance, $\\ell^1$ with the norm $\\norm{x}_1 + \\norm{x}_2$ is strictly convex.\nIndeed, equality in the triangle inequality forces equality for $\\norm{\\cdot}_2$.\nBut this space is not reflexive, so it is not uniformly convex by the [[Milman–Pettis theorem|AJ5RG9XQYZ]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "GDTKW9KPH8",
    "kind": "definition",
    "msc": [
      "46B20"
    ],
    "name": "Strictly convex space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Strictly_convex_space"
      }
    ],
    "tag": "RQDZX250B2",
    "uses": [
      "03J5RAKB30"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RQDZX250B2/",
  "links": [
    {
      "field": "defines",
      "target": "GDTKW9KPH8"
    },
    {
      "field": "uses",
      "target": "03J5RAKB30"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RQDZX250B2.md",
  "msc": [
    "46B20"
  ],
  "name": "Strictly convex space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RQD/RQDZX250B2.md",
  "status": null,
  "tag": "RQDZX250B2",
  "uri": "m2://RQDZX250B2",
  "url": "https://math2.tamnd.com/wiki/strictly-convex-space-definition/",
  "writing": {
    "tag": "RQDZX250B2",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}