If a quadratic form is positive definite on a convex set, is it convex on that set? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:27:32Z http://mathoverflow.net/feeds/question/10216 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10216/if-a-quadratic-form-is-positive-definite-on-a-convex-set-is-it-convex-on-that-se If a quadratic form is positive definite on a convex set, is it convex on that set? fuzzytron 2009-12-31T02:33:45Z 2009-12-31T16:14:03Z <p>Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A x \geq 0$ for all $x \in \mathbb{R}^n$.</p> <p>Now suppose we have a convex subset $\Phi$ of $\mathbb{R}^n$ such that $x \in \Phi$ implies $x^T A x \geq 0$. Is $x^T A x$ a convex function on $\Phi$ (even if $A$ is not positive definite)? Of course, the answer in general is "no," but we can still ask about the most inclusive conditions under which convexity holds for a given $A$ and $\Phi$. In particular I'm interested in the question:</p> <p><strong>Suppose we have a quadratic form $Q:\mathbb{R}^{n \times n} \rightarrow \mathbb{R}$. What is the <em>weakest</em> condition on $Q$ that guarantees it will be convex when restricted to the set of positive semidefinite matrices?</strong></p> http://mathoverflow.net/questions/10216/if-a-quadratic-form-is-positive-definite-on-a-convex-set-is-it-convex-on-that-se/10218#10218 Answer by fedja for If a quadratic form is positive definite on a convex set, is it convex on that set? fedja 2009-12-31T02:52:41Z 2009-12-31T03:23:58Z <p>$x^2-y^2$ is positive on $[2,3]\times [-1,1]$ but not convex there. This creates problems for any convex sets not containing the origin. You are, probably, after something else not so obviously false. Why don't you just tell us what it is?</p> <p>Edit: Even then it is false: just take $B_{11}B_{22}$. By the way, for a pure quadratic form, convexity on an open set and convexity on the entire space are the same thing. </p> http://mathoverflow.net/questions/10216/if-a-quadratic-form-is-positive-definite-on-a-convex-set-is-it-convex-on-that-se/10226#10226 Answer by Harald Hanche-Olsen for If a quadratic form is positive definite on a convex set, is it convex on that set? Harald Hanche-Olsen 2009-12-31T04:06:43Z 2009-12-31T04:06:43Z <p>The answer to the edited question is no. Let $Q\colon\Phi\to\mathbb{R}$ be the quadratic form $Q(B)=B_{11}B_{22}$. Clearly $Q\gt0$ on the set $\Phi$ of positive definite matrices. Equally clearly, this function is concave on the subset <code>$\{B\in\Phi\colon B_{11}+B_{22}=1\}$</code>.</p>