Continuous, midpoint (strictly) quasi-concave function is (strictly) quasi-concave?

Question

It is known that Midpoint-Convex and Continuous Implies Convex.

I am wondering can midpoint quasi-concavity and continuity implies quasi-concavity? If not, what conditions are required instead?

Answer 1

Yes, midpoint quasi-concavity and continuity imply quasi-concavity. Consider an upper level set $E(t) = \{x : f(x)\ge t\}$ of a midpoint quasi-concave and continuous function. For any two points $a,b\in E(t)$, their midpoint $(a+b)/2$ is also in $E(t)$. But then the points at $1/4$ and $3/4$ of the distance from $a$ to $b$ are also in $E(t)$, namely $$ \frac34a+\frac14b = \frac12\left(a+\frac{a+b}{2}\right) $$ and $$ \frac14a+\frac34b = \frac12\left(b+\frac{a+b}{2}\right) $$ Continuing this process of averaging, we obtain a dense subset of the line segment $[a,b]$ (the dyadic rationals are dense). But $E(t)$ is closed by the continuity of $f$. Hence, $[a,b]\subset E(t)$, meaning $E(t)$ is convex.

Strict quasi-concavity

If $f$ is strictly midpoint quasi-concave, then the previous argument shows that $f(c) > \min(f(a), f(b))$ for all dyadic points between $a,b$. But any point $x\in (a,b)$ falls between two dyadic points $c_1,c_2$, hence $$ f(x) \ge \min(f(c_1), f(c_2)) > \min(f(a), f(b)) $$ proving the strict quasi-concavity of $f$.

Answer 2

see the article below, called

'Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy' http://www.tandfonline.com/doi/abs/10.1080/02331939208843791?journalCode=gopt20 It discusses the relation between midpoint quasi convexity, strict midpoint quasi convexity and quasi-convexity, and quasi concavity. (and midpoint quasi concavity and quasi concavity, I believe)

It also discusses the connections between (proper) midpoint convexity (jensen convex) and the quasi-convexity to strict quasi convexity; and vice versa (that a strictly midpoint quasi convex function, that is quasi convex, is strictly quasi convex as well, and a strictly quasi convex function that is midpoint quasi convex, is quasi convex).

At least if F is Jensen (midpoint) convex. Jensen (midpoint) convex (F(x/2+y/2)<=F(x)/2+F(y)/2, the non-continuous analogue of convexity 'proper) entails both strict quasi midpoint convexity, and quasi midpoint convex, I believe; and so the two notions, two notions, strict quasi convexity and quasi convexity become equivalent under it.

And presumably but I don not know for sure, that the same would hold, under 'jensen midpoint concavity'F(x/2+y/2)=>F(x)/2+F(y)/2). strict quasi concavity and quasi concavity are equivalent, because it entails the midpoint quasi concave analogues. The paper also discusses the conditions under which a midpoint convex function that is strictly quasi convex (or quasi convex) is just convex. Concavity is discussed there is as well.