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  "body": "\n## Definition\n\nLet $K \\subseteq \\R^2$ be a [[convex body|437FSG3419]] with [[support function|1H92CYA4ZC]] $h_K$, and let $w \u003e 0$.\nFor a unit vector $u$, the *width* of $K$ in the direction $u$ is $w_K(u) = h_K(u) + h_K(-u)$.\nIt is the distance between the two [[supporting lines|K7N565NKCV]] of $K$ orthogonal to $u$.\nThe body $K$ has *constant width* $w$ if $w_K(u) = w$ for every unit vector $u$.\nA *curve of constant width* $w$ is the [[boundary|05D5EGMFYV]] of a plane convex body of constant width $w$.\n\n## Remarks\n\nExample: the circle of radius $r$ centred at $c$ bounds the disc with $h(u) = c \\cdot u + r$, so $w(u) = 2r$ for all $u$, and the circle has constant width $2r$.\nNon-example: the square $[0, 1]^2$ has width $1$ in the direction $(1, 0)$ and width $\\sqrt{2}$ in the direction $(1, 1)/\\sqrt{2}$, so its boundary is not a curve of constant width.\nA body of constant width $w$ has diameter $w$.\nThe same definition in $\\R^n$ gives bodies of constant width.\n",
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