A toy model for the t-section problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:14:06Z http://mathoverflow.net/feeds/question/66613 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66613/a-toy-model-for-the-t-section-problem A toy model for the t-section problem fedja 2011-06-01T01:43:36Z 2011-06-05T17:53:54Z <p>Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H$ is some prescribed function (small, smooth, vanishing near the endpoints to any order you wish, etc.). Is it always possible or there are some non-obvious hidden restrictions? </p> <p><img src="http://www.artofproblemsolving.com/Forum/latexrender/pictures/b/4/6/b46dbb397d96f17cce1e421bfaf29b9a80d26c81.png" alt="alt text"></p> <p>The question comes from the infamous t-section problem (if you know the areas of all sections of a symmetric convex body by the hyperplanes at some fixed small distance $t$ from the origin (so small that all sections are non-empty), can you recover the body?). The problem is open even on the plane. I do not say that this toy question is directly relevant here but an answer to it will certainly make a few things clearer for me.</p> http://mathoverflow.net/questions/66613/a-toy-model-for-the-t-section-problem/66976#66976 Answer by Douglas Zare for A toy model for the t-section problem Douglas Zare 2011-06-05T16:59:21Z 2011-06-05T17:53:54Z <p>There are restrictions.</p> <p>At most points, $S'(x)$ is the length of the right leg minus the length of the left leg of the curvilinear triangle, perhaps with exceptions on a null set where there are tangencies. If $S(0)=S(1)=0$ then the lengths of these legs are at most $\sqrt{2}(1-x)$ and $\sqrt{2}x$. For almost all $0\le x \le 1$, $S'(x)$ satisfies $-\sqrt{2}\le -\sqrt{2} x \lt S'(x) \lt \sqrt(2) (1-x) \le \sqrt{2}$. This is an extra condition on $H$ which rules out some smooth small functions which have large derivatives near some points, such as $10^6 \exp(-1/(x (1-x))^2)$ for $0\lt x \lt 1$, which has a derivative of $1.132$ at $x=0.436$ although the value of the function is small.</p>