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Abstract for Laurent Alonso and Rapha\"el Cerf,
The three dimensional polyominoes of minimal area

The set of the three dimensional polyominoes of minimal
area and of volume~$n$ contains a polyomino which is the
union of a quasicube $j\times (j+\delta)\times (j+\theta)$,
$\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$,
$\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally
associated to the unique decomposition of
$n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal
quasicube, a maximal quasisquare and a bar.  For $n$ a quasicube
plus a quasisquare, or a quasicube minus one, the minimal
polyominoes are reduced to these shapes.  The minimal area is
explicitly computed and yields a discrete isoperimetric inequality.
These variational problems are the key for finding the path of
escape from the metastable state for the three dimensional Ising model
at very low temperatures. The results and proofs are illustrated by
a lot of pictures.

\bye

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