Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x\_1,x\_2,x\_3,x\_4\}$ such that the points in $B$ are in general position (no three points on a line). 


Question 1: Is it true that $|A+B|\geq (k+1)^2$?


Question 2: Is it true that $|\cap\_{i}(A+x\_i)|\leq (k-1)^2$?


Question 3: If either of 1-2 are true, is it the case that one of the extremal
sets $A$ is a discrete square? 



Bounding the size of certain sumsets in the plane
As shown by Gardner and Gronchi ("[A Brunn-Minkowski Inequality for the Integer Lattice](http://www.ams.org/journals/tran/2001-353-10/S0002-9947-01-02763-5/S0002-9947-01-02763-5.pdf)", equality (8) / Theorem 6.6), if $A,B\subset\mathbb R^n$ are finite sets such that $B$ has full dimension, then
 $$ |A+B|^{1/n} ≥ |A|^{1/n} + \frac1{(n!)^{1/n}}\, (|B|−n)^{1/n}. $$
(This estimate should be viewed as a real-world analogue of the ideal-world "discrete [Brunn-Minkowski inequality](https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem)".)


In your situation ($n=2,\ |A|=k^2,\ |B|=4$) this yields
 $$ |A+B|^{1/2} \ge k + 1, $$
answering in the affirmative your first question.


For the second question, observe that letting $I:=\cap\_i(A+x\_i)$, we have $I-B\subseteq A$. Hence, applying the Gardner-Gronchi inequality once again,
 $$ k = |A|^{1/2} \ge |I-B|^{1/2} \ge |I|^{1/2} + 1, $$
which gives an affirmative answer to your second question.


