I believe your question (and many other variations) are answered <a href="http://www.springerlink.com/content/aw6wx84015357770/">here</a>. Also see <a href="http://www.renyi.hu/~barany/cikkek/psurveyAMS.pdf">this</a> survey by I. Barany about extremal problems for convex lattice polytopes. 

Actually the problem for the convex polygon inside the square was considered earlier than that. See <a href="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHS-4D0YB4K-9&_user=1010281&_coverDate=02%2F28%2F1995&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050264&_version=1&_urlVersion=0&_userid=1010281&md5=aeb538fb3d97eae79dfddcc646bba7f3">here</a> for an article that seems to be written more in the lines of your question.