  In this paper we discuss a simple relation, which was previously missed,
between the high co-dimensional isoperimetric problem of finding a filling with
small volume to a given cycle, and extinction estimates for singular, high
co-dimensional, mean curvature flow. The utility of this viewpoint is first
exemplified by two results which, once casted in the light of this relation,
are almost self evident. The first is a genuine, 5-lines proof, for the
isoperimetric inequality for $k$-cycles in $\mathbb{R}^n$, with a constant
differing from the optimal constant by a factor of only $\sqrt{k}$, as opposed
to a factor of $k^k$ produced by all of the other soft methods (like
Michael-Simon's or Gromov's). The second is a 3-lines proof of a lower bound
for extinction for arbitrary co-dimensional, singular, mean curvature flows
starting from cycles, generalizing the main result of a paper of Giga and
Yama-uchi. We then turn to use the above mentioned relation to prove a bound on
the parabolic Hausdorff measure of the space time track of high co-dimensional,
singular, mean curvature flow starting from a cycle, in terms of the mass of
that cycle. This bound is also reminiscent of a Michael-Simon Isoperimetric
inequality. To prove it, we are lead to study the geometric measure theory of
Euclidean rectifiable sets in parabolic space, and prove a co-area formula in
that setting. This formula, the proof of which occupies most this paper, may be
of independent interest
