L. Alonso, R. Cerf
The three dimensional polyominoes of minimal area
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ABSTRACT.  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.