Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$ be a continuously twice differentiable function.
I am considering the following condition (1) on $f$: For any $x,x'\in\mathbb R^n$,
If it is either $f(x,x')>f(x,x)$ or $f(x',x)>f(x,x)$, we should have $f(x',x')\geq\max\{f(x,x'),f(x',x)\}$.
Equivalently,
$$\tag{1}\max\{f(x,x'),f(x',x)\}>f(x,x)\Rightarrow f(x',x')\geq\max\{f(x,x'),f(x',x)\},~\forall x,x'\in\mathbb R^n.$$
My question is : Is there an equivalent condition to (1)?
One possibility is that the monotonicity of partial derivatives is related to some of the inequalities in (1).
More specifically, let the following inequalities be true for all $x,x'\in\mathbb R^n$ such that $x\neq x'$: $$(x'-x)(D_xf(x,x')-D_xf(x,x))>0~\textrm{and}$$ $$(x'-x)(D_yf(x',x)-D_yf(x,x))>0~\textrm.$$ According to my previous question (link) this monotonicity implies the following: $$f(x,x')>f(x,x)\Rightarrow f(x',x')>f(x',x).$$
We have the relationship between $f(x',x')-f(x',x)$ and $f(x,x')-f(x,x)$. But what is missing to answer (1) is the relationship between $f(x',x')-f(x,x')$ and $f(x,x')-f(x,x)$, that I cannot find from my previous question. Any condition that reveals the relationship between $f(x',x')-f(x,x')$ and $f(x,x')-f(x,x)$?
Any comment and answer will be highly appreciated.
An equivalent condition is: For all $x,x' \in \mathbb{R}^n$ $$ \iff \left[ \begin{array}{ll} \max\{f(x',x'),f(x,x) \} \ge \max\{f(x,x'),f(x',x) \} > \min\{f(x',x'),f(x,x) \} \\ \min\{f(x',x'),f(x,x) \} \ge \max\{f(x,x'),f(x',x) \} \end{array} \right. $$ where the right bracket means "either ..or".
I don't know whether these conditions have a name.