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[[Barbier's theorem]] asserts that the [[perimeter]] of any curve of constant width is equal to the width multiplied by <math>\pi</math>. As a special case, this formula agrees with the standard formula <math>\pi d</math> for the perimeter of a circle given its diameter.{{r|lay|barbier}} By the [[isoperimetric inequality]] and Barbier's theorem, the circle has the maximum area of any curve of given constant width. The [[Blaschke–Lebesgue theorem]] says that the Reuleaux triangle has the least area of any convex curve of given constant width.{{r|gruber}} Every proper superset of a body of constant width has strictly greater diameter, and every Euclidean set with this property is a body of constant width. In particular, it is not possible for one body of constant width to be a subset of a different body with the same constant width.{{r|eggleston|jessen}} Every curve of constant width can be approximated arbitrarily closely by a piecewise circular curve or by an [[analytic curve]] of the same constant width.{{r|wegner77}}
A [[vertex (curve)|vertex of a smooth curve]] is a point where its curvature is a local maximum or minimum; for a circular arc, all points are vertices, but non-circular curves may have a finite discrete set of vertices. For a curve that is not smooth, the points where it is not smooth can also be considered as vertices, of infinite curvature. For a curve of constant width, each vertex of locally minimum curvature is paired with a vertex of locally maximum curvature, opposite it on a diameter of the curve, and there must be at least six vertices. This stands in contrast to the [[four-vertex theorem]], according to which every simple closed smooth curve in the plane has at least four vertices. Some curves, such as ellipses, have exactly four vertices, but this is not possible for a curve of constant width.{{r|martinez|ctb}} Because local minima of curvature are opposite local maxima of curvature, the only curves of constant width with [[central symmetry]] are the circles, for which the curvature is the same at all points.{{r|mmo}} For every curve of constant width, the [[Smallest-circle problem|minimum enclosing circle]] of the curve and the largest circle that it contains are concentric, and the average of their diameters is the width of the curve. These two circles
A convex body has constant width if and only if the Minkowski sum of the body and its central reflection is a circular disk; if so, the width of the body is the radius of the disk.{{r|mmo|chakerian}}
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