TITLE: For quasi-convex functions, is $f (y)\leq\max(f(x_1),\ldots,f(x_n)) $ when $y = \frac{1}{n}\sum x_i$?
QUESTION [2 upvotes]: Definition :
The function $f:D \to \mathbb{R}$ where $D$ is a convex set in $\mathbb{R}^m$ is said quasi-convex on $D$ if :
$$
f(tx + (1-t)y) \leq \max (f(x),f(y))\qquad x,y\in D~\text{and}~ t\in[0,1]
$$

Let $x_1,\ldots,x_n,y\in D $ such that :
$$
y=\frac {1}{n}\sum_{k=1}^{n}{x_k}
$$
Can we  say that :
$$
f (y)\leq\max(f(x_1),\ldots,f(x_n))
$$
Thank you for all comments and helping.

REPLY [3 votes]: $$f(y)=f\Big(\frac{1}{n}x_1+\frac{n-1}{n}\frac{1}{n-1}\sum_{k=2}^{n}x_k\Big)$$
$$\leq \max\Big\{f(x_1),f\Big(\frac{1}{n-1}\sum_{k=2}^{n}x_k\Big)\Big\}$$
and from here you can use induction on $n$.
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