Popoviciu's inequality states that for convex $f$ and numbers $x_1,x_2,x_3$, we have $f(x_1)+f(x_2) + f(x_3) + 3\cdot f(\frac{x_1+x_2+x_3}3) \geq 2\cdot f(\frac{x_1+x_2}2)+2\cdot f(\frac{x_1+x_3}2)+2\cdot f(\frac{x_2+x_3}2)$.

It can be proven just by using Karamata's inequality and arguing that the sequence of $6$ numbers on the LHS majorizes the sequence on the RHS. Some modest generalizations were proposed, where we would have $n$ numbers, LHS would have still a sum of the function for each single argument + the mean and the RHS would have average of all possible $k$ of $n$ tuples.

Can we go full measure here with all possible combinations of numbers? That is, $\sum f(x_i) - 2 \sum_{i_1 < i_2 } f(\frac{x_{i_1}+x_{i_2}}{2})+ 3 \sum_{i_1 < i_2 < i_3} f(\frac{x_{i_1}+x_{i_2}+x_{i_3}}{3}) + ... + (-1)^{n+1} nf(\frac{x_{i_1}+...+x_{i_n}}{n}) \geq 0$

I would like to tackle this with Karamata's inequality and argue that the sequence for positive numbers majorizes the sequence for negative numbers. However already for $n=3$ the only way to show the majorization is with case work. So I would like to know if there are some smarter ways of doing so. Or maybe you know some obvious reasons for which the whole inequality just cannot be true.