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!

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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!

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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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