  In two dimensional digital geometry, two lattice points are 4-connected
(resp. 8-connected) if their Euclidean distance is at most one (resp.
$\sqrt{2}$). A set $S \subset Z^2$ is 4-connected (resp. 8-connected) if for
all pair of points $p_1, p_2$ in $S$ there is a path connecting $p_1$ to $p_2$
such that every edge consists of a 4-connected (resp. 8-connected) pair of
points. The original definition of digital convexity which states that a set $S
\subset Z^d$ is digital convex if $\conv(S) \cap Z^d= S$, where $\conv(S)$
denotes the convex hull of $S$ does not guarantee connectivity. However,
multiple algorithms assume connectivity. In this paper, we show that in two
dimensional space, any digital convex set $S$ of $n$ points is unimodularly
equivalent to a 8-connected digital convex set $C$. In fact, the resulting
digital convex set $C$ is 4-connected except for at most one point which is
8-connected to the rest of the set. The matrix of $SL_2(Z)$ defining the affine
isomorphism of $Z^2$ between the two unimodularly equivalent lattice polytopes
$S$ and $C$ can be computed in roughly $O(n)$ time. We also show that no
similar result is possible in higher dimension.
