2 corrected minor typo.

Actually, @Yoav's answer is more relevant than he thinks. In this paper:

MR1837217 (2002g:52011) Gardner, R. J.(1-WWA); Gronchi, P.(I-CNR-GA) A Brunn-Minkowski inequality for the integer lattice. (English summary) Trans. Amer. Math. Soc. 353 (2001), no. 10, 3995–4024 (electronic).

The authors prove just what they say they prove. The form of the inequality is a little different, but for example, using @Yoav's computation (which I am loath to copy and paste, he might want to undelete his answer), you get the following: $|(K+L)/2|^2 \geq |K|(|L| - n)/n!$ in dimension $n$ (assuming $L$ is full-dimensionalfull-dimensional). They have better inequalities in $2$ dimensions, but you should just read the (very well written) paper.

1

Actually, @Yoav's answer is more relevant than he thinks. In this paper:

MR1837217 (2002g:52011) Gardner, R. J.(1-WWA); Gronchi, P.(I-CNR-GA) A Brunn-Minkowski inequality for the integer lattice. (English summary) Trans. Amer. Math. Soc. 353 (2001), no. 10, 3995–4024 (electronic).

The authors prove just what they say they prove. The form of the inequality is a little different, but for example, using @Yoav's computation (which I am loath to copy and paste, he might want to undelete his answer), you get the following: $|(K+L)/2|^2 \geq |K|(|L| - n)/n!$ in dimension $n$ (assuming $L$ is full-dimensional. They have better inequalities in $2$ dimensions, but you should just read the (very well written) paper.