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Jan 17, 2021 at 15:07 comment added Charles Pehlivanian Yes, thanks, this does it. I haven't looked at counterexamples enough apparently. The property follows trivially if I also assume subadditivity of $f$ (or 2-subhomogeneity, since $f$ is convex), but then that and the first bullet are sufficient, no partial deriv constraints needed. I was looking to avoid a subadditivity assumption...
Jan 17, 2021 at 14:27 vote accept Charles Pehlivanian
Jan 17, 2021 at 13:26 history edited Pietro Majer CC BY-SA 4.0
added 10 characters in body
Jan 17, 2021 at 13:21 history answered Pietro Majer CC BY-SA 4.0