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Sep 16, 2020 at 22:47 comment added fedja It is a bit difficult because the least convex (down) majorant of $\log\frac{1+x+y}{1+y}$ in the first quadrant is exactly $\log(1+x)$, and you are essentially asking about the least convex majorant of the sum of such functions in the domain $x\ge0$, $y\ge Bx$ with general matrix $B$ with positive entries so you'll have to incorporate your $\beta$ into the objective function in some essential way to get anything useful. Well, you, probably, know that already :-)
Sep 16, 2020 at 19:26 history edited YCor CC BY-SA 4.0
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Sep 16, 2020 at 16:00 history asked Math_Y CC BY-SA 4.0