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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!

  • 1
    $\begingroup$ Can you show us your proof for n = 2? $\endgroup$ – Mark L. Stone Jun 13 '18 at 22:08
  • $\begingroup$ 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. $\endgroup$ – Eric Zaslow Jun 13 '18 at 23:12
  • $\begingroup$ Logarithmic concavity of the function is the key, isn't it? $\endgroup$ – fedja Jun 14 '18 at 1:57
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    $\begingroup$ This comment, fedja, seems to presuppose that I know the answer. I don't. If you do, please elaborate. $\endgroup$ – Eric Zaslow Jun 14 '18 at 4:15
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    $\begingroup$ 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). $\endgroup$ – 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.


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