Let $(u\_1, u\_2, u\_3, u\_4)$ and $(v\_1, v\_2, v\_3, v\_4)$ be vectors in $\mathbb R\_+^4$. Is the following inequality true? 


\begin{align\*}
\left(\sum\_{{[4] \choose 3}} \sqrt{u\_i u\_j u\_k}\right)^{2/3} + \left(\sum\_{{[4] \choose 3}} \sqrt{v\_i v\_j v\_k}\right)^{2/3} \leq \left(\sum\_{{[4] \choose 3}} \sqrt{(v\_i+u\_i) (v\_j+u\_j) (v\_k+u\_k)}\right)^{2/3}
\end{align\*}
Here $\sum\_{{[4] \choose 3}}$ refers to $\sum\_{1\leq i < j < k \leq 4}$. 


I ran a million Matlab simulations for random vectors and it did not yield any counterexample. 


Note: Previously asked on MSE (<https://math.stackexchange.com/questions/2647608/a-minkowski-like-inequality-for-symmetric-sums>) 



A (reverse)-Minkowski type inequality for symmetric sums
Rewrite the inequality in question as 
\begin{equation\*}
f(u+v)\le f(u)+f(v) 
\end{equation\*}
for $u,v$ in $\mathbb R\_+^4$, 
where 
\begin{equation\*}
 f(u):=-\left(\left(\frac{1}{\sqrt{u\_1}}+\frac{1}{\sqrt{u\_2}}+\frac{1}{\sqrt{u\_3}}
 +\frac{1}{\sqrt{u\_4}}\right) \sqrt{u\_1 u\_2 u\_3 u\_4}\right)^{2/3}. 
\end{equation\*}
Note that the function $f$ is positive homogeneous: $f(tu)=tf(u)$ for $t\ge0$. So, $f(u+v)=2f(\frac{u+v}2)$. 


It remains to notice that $f$ is convex. Indeed, the determinant of the Hessian matrix 
\begin{equation}
 M:=\Big(\frac{\partial^2 f}{\partial u\_i\partial u\_j}\Big)\_{i,j=1}^4
\end{equation}
is $0$, and the principal minors of $M$ are manifestly positive, after some algebraic simplifications. (That the determinant of $M$ is $0$ can be shown either by direct calculations or by recalling that $f$ is positive homogeneous and hence $\frac{d^2}{dt^2}\,f(tu)=0$ for $t>0$.)


Dealing with the determinants of the matrices $\Big(\frac{\partial^2 f}{\partial u\_i\partial u\_j}\Big)\_{i,j=1}^k$ for $k=1,2,3$, in view of the positive homogeneity, we may assume without loss of generality that $u\_4=1$. 


Details of the calculations can be seen in the [the Mathematica notebook](https://my.pcloud.com/publink/show?code=XZ6QYR7ZEfYGuaIKnuuUKFtgIHSSkRJU2Gy0) and/or [its pdf image](https://my.pcloud.com/publink/show?code=XZUQYR7ZvyvpE9vzMvVphBLqG6p6aLzM9eck) . 


