It does work on irregular polygons, provided all their sides are the same length and they're convex. For example, it does work for the house-shaped pentagon I described above: for any point within the house shape, the sum of the perpendicular distances from all five sides is a constant.

... *thinks* i was thinking of the dist between the point and the base... not the sides
oops.
still not 100% convinced... need to get a piece of paper!
is it the sum of ALL the sides then?
that would help :)