Naive tomography question - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:06:54Z http://mathoverflow.net/feeds/question/84918 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84918/naive-tomography-question Naive tomography question Vincent 2012-01-04T23:38:16Z 2012-01-05T00:59:01Z <p>Given a vectorspace $V$ (over a field $F$) with a specified basis $b_1, \ldots b_n$ and a set $S \subset V$ with two properties:</p> <p>1) $S$ is a union of lines through the origin (so for all $s \in S$ we have that $Fs \subset S$) </p> <p>2) For every affine line line parallel to one of the basis vectors intersects $S$ in exactly one point. (So for every $i \in {1, \ldots n}$ and every $v \in V$ we have $|(v + Fb_i) \cap S| = 1$.)</p> <p>Is $S$ necessarily a codimension 1 linear subspace or are there other examples?</p> http://mathoverflow.net/questions/84918/naive-tomography-question/84923#84923 Answer by Robert Israel for Naive tomography question Robert Israel 2012-01-05T00:08:02Z 2012-01-05T00:59:01Z <p>In ${\mathbb R}^3$ with the standard basis try the surface $x^3 + y^3 + z^3 = 0$.</p> <p><img src="http://www.math.ubc.ca/~israel/problems/vincent.jpg" alt="alt text"></p> http://mathoverflow.net/questions/84918/naive-tomography-question/84924#84924 Answer by Yoav Kallus for Naive tomography question Yoav Kallus 2012-01-05T00:10:25Z 2012-01-05T00:10:25Z <p>I think this is a counterexample in $\mathbb{R}^3$:</p> <p>Start with the plane with normal direction $(1,1,1)$. Tilt each radial ray of that plane by an angle $\alpha = \epsilon \sin 3\phi$ towards $(1,1,1)$ (or away if negative), where $\phi$ is the angle the ray makes with the ray through, say, $(1,-1,0)$. For small $\epsilon$, the resulting surface is still a graph of a single valued function $x=f_1(y,z)$ or $y=f_2(z,x)$ or $z=f_3(x,y)$.</p>