Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 35 (1994), No. 1, 37-43.

Maximum Convex Hulls of Connected Systems of Segments and of Polyominoes

K\'aroly Bezdek, Peter Bra\ss, Heiko Harborth

Abstract.

In this note we prove the following two isoperimetric-type theorems.
The $d$-dimensional volume of the convex
hull of any connected system of finitely
many segments in ${\bbf R}^d$
with total length 1 which are parallel to
the standard co-ordinate axes is at most ${1\over d^d d!}$.
Moreover, the area of the convex hull of
any facet-to-facet connected system of $n$ unit squares in ${\bbf R}^2$,
that is, a polyomino with $n$ cells
[7], is at most $n +
\left\lfloor{n-1\over 2}\right\rfloor\left\lfloor{n\over 2}\right\rfloor$.