Proving that a specific function is quasiconvex - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T17:16:24Z http://mathoverflow.net/feeds/question/105474 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105474/proving-that-a-specific-function-is-quasiconvex Proving that a specific function is quasiconvex Josh 2012-08-25T17:28:32Z 2012-08-27T16:54:52Z <p>Hello all,</p> <p>Assume we have a sequence of <strong>quasiconcave</strong> functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function $f_{i,j}(X)$. </p> <p>Assuming that $F\succ0$ (positive definite for all $X$), I want to prove (or disprove) that the function $g(X)=a^TF^{-1}a$, where $a\in\mathbb{R}_+^{n\times 1}$, is quasiconvex. </p> <p>Someone have any idea?</p> <p>Thank you!</p> <hr> <p>Correction: $f_{i,j}(X)$ are quasiconcave and not quasiconvex. Credit to Robert.</p> http://mathoverflow.net/questions/105474/proving-that-a-specific-function-is-quasiconvex/105638#105638 Answer by Robert Israel for Proving that a specific function is quasiconvex Robert Israel 2012-08-27T16:54:52Z 2012-08-27T16:54:52Z <p>It's not true.</p> <p>Consider the $2 \times 2$ matrix $$F(X) = \pmatrix{f(X) &amp; 0\cr 0 &amp; f(X-2)\cr}$$ where $f$ is an even function, everywhere $> 0$, and decreasing on $[0,\infty)$. Take $a = (1,1)^T$. Then $g(X) = a^T F(X)^{-1} a = 1/f(X) + 1/f(X-2)$. In particular $g(0) = g(2) = 1/f(0) + 1/f(2)$ while $g(1) = 2/f(1)$.<br> If $2/f(1) > 1/f(0) + 1/f(2)$, $g$ is not quasiconvex. For example, we could take $f(0) = 4$, $f(1) = 2$, $f(2) = 1$.</p>