Sampling points on the edges and corners of a hybercube via LHS I am looking for some source that supports my intuition, that the Latin Hypercube Design rarely (or never?) samples points from the surface.
A point lies on the surface of a hypercube if at least one of the $i$ variables is equal to the upper- or lower limit defined for the corresponding $i$th variable.
$P(X_i  = a_{i} \ or \ X_i = b_{i})$
Imagining trials for a constant number of samples and dimensions, are there differences between the different LHS implementations (e.g. from the R-package lhs) on how often or if ever points from the surface would be sampled?

It would help to be specific about what you mean by "the edge".
For example, if one of the lhs parameters is $X_1$~Uniform(a, b), then $P(X_1 = a) = 0$ and $P(X_1 = b) = 0$.
If you mean, $P(X_1 \ge a \ \ and\ \  X_1 \le a + \epsilon)$ then you can estimate this in a straightforward way based on the distributions in the lhs parameters.
No matter which implementation of lhs construction you use, you will always have one point on the left and right "edge" of every variable.  The lhs is required to have one observation in each parameter quantile or bin.
