We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, i.e the all coordinates positive quadrant of $R^n$, and obviously it is connected as we only consider positive quadrant.

My question is: whether finite intersection of semi algebraic sets defined by above inequalities is still connected or not?

Personally, I believe it is connected by checking several examples by hand, but still have not idea about how to give general proof. Thanks a lot.

  • 2
    $\begingroup$ If I understand the question correctly (I'm not sure), why don't you take log of your coordinate system? You will get convex polytops. $\endgroup$
    – Uri Bader
    May 4, 2016 at 7:00


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