# Quasi-concavity of A Minimized Quasi-concave Function

Let function $$F(x,y,z)$$ defined on $$[0,1]\times[0,1]\times[0,1]$$ be increasing in $$x$$ and $$y$$. By increasing I mean $$\frac{\partial F}{\partial x}\geq0$$ and $$\frac{\partial F}{\partial y}\geq0$$.

By quasi-concave I mean $$F(x',y',z)\geq F(x,y,z)\Rightarrow[\frac{\partial F}{\partial x},\ \frac{\partial F}{\partial y}][x'-x,\ y'-y ]^T\geq0.$$

Define $$H(x,y)=\min\limits_{z\in[x,1]}F(x,y,z).$$

My question: Is function $$H(x,y)$$ quasi-concave in $$x$$ and $$y$$? Why?

EDIT 1: I think $$H(x,y)$$ is still increasing in $$x$$ and $$y$$, because for any $$x'\geq x$$ and $$y'\geq y$$, we have $$\min\limits_{z\in[x',1]}F(x',y,z)\geq \min\limits_{z\in[x,1]}F(x,y,z)$$ and $$\min\limits_{z\in[x,1]}F(x,y',z)\geq \min\limits_{z\in[x,1]}F(x,y,z)$$. Is it right?

EDIT 2: Correction of the quasi-concavity.

• Well. If $F$ is increasing I can also say it's quasi convex right? I believe by increasing you mean increasing in any variable if others are fixed. For e.g. increasing in $z$ if $x$ and $y$ are fixed. Jun 25, 2020 at 11:52
• @Shiv Tavker Yes, but I'm confused about the minimized function. I want to see if function H is quasi-concave or not...
– HXW
Jun 25, 2020 at 12:08
• Isn't the argmin just $z^* = x$, given the way you are setting up the problem?
– user762914
Jun 25, 2020 at 12:26
• @Renard But function F is not necessarily increasing in z...
– HXW
Jun 25, 2020 at 13:26
• @Huaixin Now the First inequality in Edit 1 may not be correct. Jun 25, 2020 at 14:07

An increasing function is only necessarily quasiconcave if it is a map from $$\mathbb{R}$$ to $$\mathbb{R}$$. The function $$x^2+y^2+z^2$$ is strictly convex on $$[0,1]^3$$, and increasing for any meaningful definition of increasing.

• True. I guess calling quasi convex is safer? Jun 25, 2020 at 12:31
• Thanks! I corrected my formulation.
– HXW
Jun 25, 2020 at 13:52

Since, $$F(x, y, z)$$ is increasing, $$\ H(x, y) = \min_{z\in[x, 1]}F(x, y, z) = F(x, y, x)$$ Can you finish it now?

• F is increasing in x and y, but it is not necessarily increasing in z.
– HXW
Jun 25, 2020 at 13:23
• Oh you edited the question! Jun 25, 2020 at 14:03
• I just corrected the formulation, sorry...
– HXW
Jun 25, 2020 at 14:06