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corrected minor typo.

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edited May 15 2012 at 0:00

Igor Rivin
31.2k●3●37●71

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.

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answered May 14 2012 at 23:15

Igor Rivin
31.2k●3●37●71

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

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.