András Bezdek - On fat polygons and polyhedra

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András Bezdek - On fat polygons and polyhedra

	ANDRÁS BEZDEK, The Mathematical Institute of the Hungarian Academy of Science, Budapest, Auburn University, Auburn, Alabama 36849-5310, USA
	On fat polygons and polyhedra

(Joint work with Ferenc Fodor)

We investigate the problem of finding the polygons (polyhedra resp.) with n vertices and of diameter 1 which have the largest possible width w(n) (W(n) resp). We prove that $w(3)=w(4)=\frac{\sqrt3}2$ and $W(4)=W(5)= \frac 1 {\sqrt2}$ and in general $w(n) \leq \cos \frac \pi {2n}$ . In the later upper bound equality holds if n has an odd divisor greater than 1 and in this case a polygon ${\cal P}$ is extremal if and only if it has equal sides and it is inscribed in a Reuleaux polygon of constant width 1, so that the vertices of the Reuleaux polygon are also vertices of ${\cal P}$ .

eo@camel.math.ca