Curve of constant width

Builds on Convex body Support function Supporting hyperplane Boundary

Definition

Let $K \subseteq \R^2$ be a convex body with support function $h_K$ , and let $w > 0$ . For a unit vector $u$ , the width of $K$ in the direction $u$ is $w_K(u) = h_K(u) + h_K(-u)$ . It is the distance between the two supporting lines of $K$ orthogonal to $u$ . The body $K$ has constant width $w$ if $w_K(u) = w$ for every unit vector $u$ . A curve of constant width $w$ is the boundary of a plane convex body of constant width $w$ .

Remarks

Example: 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 $2r$ . Non-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. A body of constant width $w$ has diameter $w$ . The same definition in $\R^n$ gives bodies of constant width.

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Tag	`F2JX0R86WQ`
URI	`m2://F2JX0R86WQ`
Kind	definition
Writing	W0 stub
MSC	52A10
Defines	Curve of constant width
Uses	Convex body, Support function, Supporting hyperplane, Boundary

Curve of constant width

Definition

Remarks

Reading path

Further reading

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