a layer in $R^{k}$ is sum of entries of a integer point. for example $(1,5,3)$ lie on the layer $1+5+3=9$.
suppose we have a simplex whose vertices are permuted an integer point. for example a simplex with vertices $(3,2,1,4)$, $ (4,3,2,1)$, $(1,4,3,2)$, $(2,1,4,3)$ in $R^{4}$.
my question is that can we find any condition for the vertices of new simplex who can have smaller simplex in its interior(in the same layer)? for example in $ R^{4}$ there is an example in $R^{4}$ that: adding 1 to the biggest entry and subtracting 1 from another entry.
I think in higher dimension we need to change more coordinates. am I right?
it should be bigger than $\sqrt{2}$