# Is the level set of a product of affine linear functions comprised of convex curves?

Internet searches haven't helped. Can you?

Let $\, f = \prod_{i=1}^n (a_i x + b_i y + c_i).$ Is each component of $\, f^{-1}(1)$ a convex curve?

I expect so, and can prove it for $n=2,$ but I'm hopeless beyond that. Thanks!

• Can you show us your proof for n = 2? – Mark L. Stone Jun 13 '18 at 22:08
• The proof itself is probably too boring, but I thought the method might generalize: use implicit differentiation to compute $y''$, then show this is essentially a perfect square. – Eric Zaslow Jun 13 '18 at 23:12
• Logarithmic concavity of the function is the key, isn't it? – fedja Jun 14 '18 at 1:57
• This comment, fedja, seems to presuppose that I know the answer. I don't. If you do, please elaborate. – Eric Zaslow Jun 14 '18 at 4:15
• There isn't much to elaborate upon, really. Note that every component is just the boundary of a connected open set where $f>1$ and that this set is convex by the log-concavity of linear functions (you cannot change the sign of any linear factor within one connected set because it will immediately bring you to $0$ in between). – fedja Jun 14 '18 at 19:18

fedja's solution: where $f>0$, $\log(f)$ is defined, with nonpositive Hessian. Thus $\log(f)$ is concave, hence has convex superlevel sets.