For question 1, I expected some explanation that a cell receives a
negative signal from its neighbors if they are receiving a positive
signal.  Not just that cells are "connected" in some way.
The formula of the Laplacian was expected.  For part (b) you should have
drawn a simple neural net and had the inputs, outputs and weights marked.
For part (c) I generally accepted explanations that the Laplacian and
lateral inhibition are "opposites", or that the Laplacian can be used to
simulate l.i. The more detail the better.  This is all in a .ps file
by Godfried.

For q.2 there is a simple counterexample that I think everybody found.
It is a non-symmetric crossing quadrilateral.

For q.3, some people didn't specify that for a point to be on the medial
axis, not only must it have two equidistant points on the boundary, but
these points must be the closest boundary points.  The example that you
had to draw contained parabolic arcs associated with the concave vertices
(no parts of the axis touch a concave vertex).

For q.4 everybody seemed to have the right idea of infinite Minkowski
metric is, but D*(P) applies to the closest boundary position, not some
arbitrary boundary position.  The correct transofrm can be found by just
peeling off layers of the boundary.  I gave marks in part (b) even if you
didnt get part (a) correct.