Is there an ellipsoid with given outer normals? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:14:01Z http://mathoverflow.net/feeds/question/78205 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78205/is-there-an-ellipsoid-with-given-outer-normals Is there an ellipsoid with given outer normals? Elena Yudovina 2011-10-15T11:16:19Z 2011-10-15T11:16:19Z <p>Pick two points \$(x,0)\$ and \$(0,y)\$ (say \$x>0\$ and \$y>0\$). Pick a unit vector \$u = (u_1,u_2)\$, \$v = (v_1, v_2)\$, and attach one to each of the points. Provided \$u\$ and \$v\$ are "nice" (\$v\$ needs to lie "between" \$(u_1,u_2)\$ and \$(-u_1,u_2)\$ in the appropriate sense -- in their convex hull if \$u_2 > 0\$, outside of it if \$u_2 &lt; 0\$, and \$v_1 > 0\$ if \$u_2 = 0\$), there exists a unique ellipse with major axes parallel to the coordinate axes, passing through the two points, and with the unit vectors as its unit outer normals. (I think.)</p> <p>Is there a generalization of this statement to higher dimensions?</p>