Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following:

Question 1: Find sufficient conditions on $A$ so we have: $s_3(A)\geq (d+2)s_2(A) -\binom{d+2}{2}s_1(A)+\binom{d+2}{3} $?

For $d=1$ $A$ is just a bunch of integers. Then the inequality $s_3(A)\geq 3s_2(A)-3s_1(A)+1$ occurs if they form an arithmetic progression (in fact, equality happens). I did not find other examples, although admittedly I have not looked too hard (Seva pointed out in the comment that this hold for "generic" A). It is of course easy and classical that $s_2(A)\geq 2s_1(A)-1$.

A harder(?) question is:

Question 2: Find sufficient conditions on $A$ so we have: $s_3(A)\geq (d+1)s_2(A) -\binom{d+1}{2}s_1(A)+\binom{d+1}{3} $?

This inequality is weaker than the one in Question 1. For $d=1$ it reads $s_3(A)\geq 2s_2(A)-s_1(A)$. In fact, I do not know any sequence that fails this.

Any pointer to relevant, similar inequalities in the literature would be greatly appreciated.