Inequality of arithmetic and geometric means for the lattice polytopes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:41:28Z http://mathoverflow.net/feeds/question/96931 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96931/inequality-of-arithmetic-and-geometric-means-for-the-lattice-polytopes Inequality of arithmetic and geometric means for the lattice polytopes? Li Li 2012-05-14T18:04:41Z 2012-05-15T08:59:09Z <p>Let $K$,$L$,$M$ be convex lattice polytopes (so their vertices are in $\mathbb{Z}^n$) in $\mathbb{R}^n$ satisfying $K+L\subseteq M+M$ (Minkowski sum). Do we always have $$|K\cap\mathbb{Z}^n|\cdot|L\cap\mathbb{Z}^n|\le |M\cap\mathbb{Z}^n|^2 ?$$</p> <p>I looked at books/papers on Erhart polynomial and Brunn-Minkowski inequality etc., but did not find an satisfying answer. Any comment? Thanks.</p> http://mathoverflow.net/questions/96931/inequality-of-arithmetic-and-geometric-means-for-the-lattice-polytopes/96936#96936 Answer by Yoav Kallus for Inequality of arithmetic and geometric means for the lattice polytopes? Yoav Kallus 2012-05-14T19:08:44Z 2012-05-14T23:35:05Z <p><strong>Undeleted for the sake of a full record</strong></p> <p>Your inequality seems to follow easily from the Brunn-Minkowski inequality. Namely,</p> <p>$$|(K+L)/2|^2 \ge \left[|K/2|^{1/n}+|L/2|^{1/n}\right]^{2n} \ge\left[4|K/2|^{1/n}|L/2|^{1/n}\right]^{n} =|K||L|\text.$$</p> http://mathoverflow.net/questions/96931/inequality-of-arithmetic-and-geometric-means-for-the-lattice-polytopes/96954#96954 Answer by Igor Rivin for Inequality of arithmetic and geometric means for the lattice polytopes? Igor Rivin 2012-05-14T23:15:50Z 2012-05-15T00:00:33Z <p>Actually, @Yoav's answer is more relevant than he thinks. In this paper:</p> <p>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). </p> <p>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.</p> http://mathoverflow.net/questions/96931/inequality-of-arithmetic-and-geometric-means-for-the-lattice-polytopes/96956#96956 Answer by Ilya Bogdanov for Inequality of arithmetic and geometric means for the lattice polytopes? Ilya Bogdanov 2012-05-14T23:57:50Z 2012-05-14T23:57:50Z <p>This inequality does not necessarily hold, at least for $n\geq 3$. It is somehow connected with the fact that there is no Pick's formula in more than two dimensions since there exists a convex lattice polytope with a large volume but containing a small number of lattice points. </p> <p>So, for instance, for $n=3$ let $M$ be the convex hull of the points $(0,0,0)$, $(1,1,0)$, $(0,1,2k)$ and $(1,0,2k)$. Then $|M\cap {\mathbb Z}^3|=4$, but $(M+M)\cap {\mathbb Z}^3\supset \{(1,1,t):0\leq t\leq 2k\}$. So $K$ and $L$ can be chosen as vertical segments containing $k$ lattice points each.</p>