Question regard checking convexity by "restriction to any line that intersects the function domain" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:38:16Z http://mathoverflow.net/feeds/question/106174 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106174/question-regard-checking-convexity-by-restriction-to-any-line-that-intersects-th Question regard checking convexity by "restriction to any line that intersects the function domain" Josh 2012-09-02T11:35:35Z 2012-09-02T13:27:56Z <p>Hello all,</p> <p>I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain".</p> <p>In Stephen Boyd and Lieven Vandenberghe book ("Convex Optimization") they present the following example:</p> <p>"For the function $f(X) = \log\text{det}f(X)$ we can verify concavity by considering an arbitrary line, given by $X = Z+tV$ where $Z,V$ are symmetric matrices. We define $g(t) = f(Z+tV)$ and restrict $g$ to the interval of values of $t$ for which $Z+tV\succ0$." <strong>Now, without loss of generality, they assume that $t=0$ is inside this interval, i.e.</strong> $Z\succ0$. Which is fine!</p> <p>Using some algebraic manipulation, and the second order condition for concavity, they show that $g$ is concave.</p> <p>My question : Is it true to assume without loss of generality that $V\succ0$ instead of $Z\succ0$ ? After all, if we do assume that, we allways can find matrices $V$ such that $Z+tV\succ0$. Also, why we can't assume that both $Z\succ0$ and $V\succ0$. Maybe I miss something regard the "restricted to any line that intersects its domain"..</p> <p>Thank you!</p> http://mathoverflow.net/questions/106174/question-regard-checking-convexity-by-restriction-to-any-line-that-intersects-th/106177#106177 Answer by Igor Rivin for Question regard checking convexity by "restriction to any line that intersects the function domain" Igor Rivin 2012-09-02T13:27:56Z 2012-09-02T13:27:56Z <p>Think of $V$ as a tangent vector at $Z.$ you can see that, since the set of psd matrices is an open subset of the set of all symmetric matrices, there is absolutely no restriction on $V.$ On the other hand, the result (concavity of log det ) only holds in the psd cone, so $Z$ better be an element of the cone.</p>