# Problem with the vertices of a convex quadrilateral on integer lattice

I made the following observation and I am wondering if it is always true.

Let $x_1$, $x_2$, $x_3$ and $x_4$ be four positive integer points in the plane ($x_i\in\mathbb{Z^2_{\geq 0}}$) forming a convex quadrilateral and let $x=x_1+x_2+x_3+x_4$. Then it seems it is always possible to write $x$ as the nonnegative integer combination of at most three integer points inside, or on the edges, of the convex quadrilateral.

Example 1. With $x_1=(0,1)$, $x_2=(0,3)$, $x_3=(2,1)$, $x_4=(2,3)$, we have $x=(4,8)$. And we can write $x$ as $x=2(2,1)+2(0,3)$.

Example 2. With $x_1=(0,2)$, $x_2=(1,3)$, $x_3=(3,0)$, $x_4=(5,1)$, we have $x=(9,6)$. And we can write $x$ as $x=(2,1)+2(2,2)+(3,1)$.

• Is this observation true, or is there any counterexample? EDIT: Following the answer of fedja: yes it is always true, and obvious after all.
• EDIT: In higher dimension $n\geq 3$, is it possible to generalize and say that the vector $x$, made of the sum of $2n$ integer points $x_i\in\mathbb{Z^n_{\geq 0}}$, can be written as the nonnegative integer combination of at most $2n-1$ integer points inside, or on the facets, of the convex polyhedron which has vertices $x_i$.

Thank you very much!

• True, of course, because the parallelogram built on some 2 adjacent sides is contained in the quadrilateral, so we always have the representation of the kind $(x_1+x_2-x_3)+2x_3+x_4$ for some re-indexation of the vertices. However this is a typical MSE question, not an MO one. Ask such stuff there next time. – fedja Jun 28 '18 at 9:52
• @fedja Thank you very much for you answer! Sorry if it is too obvious. And in dimension n>=3, do you think it is possible to say that the sum of 2n integer points can always be written as a nonnegative integer combination of at most 2n-1 points? – B. Gimazid Jun 28 '18 at 15:48
• Caratheodory's Theorem states that if a point $x$ of ${\bf R}^d$ lies in the convex hull of a set $P$, then $x$ can be written as the convex combination of at most $d+1$ points in $P$ (copied from en.wikipedia.org/wiki/…). I wonder whether this theorem can be applied here. – Gerry Myerson Jun 28 '18 at 23:57