An extension on Jensen inequality

Question

Suppose $f(x)$ is convex in $[0,1]$, then define $$ x_{n}=\frac{1}{2 n}\left[f(0)+2 f\left(\frac{1}{n}\right)+2 f\left(\frac{2}{n}\right)+\cdots+2 f\left(\frac{n-1}{n}\right)+f(1)\right] $$ I want to know whether $x_{n}$ is decreasing with $n$, it seems to be right, but I don’t know how to prove.

Answer 1

Hopefully I have not made any mistakes. I view the above as integration with respect to the measure $\mu_n = {1 \over 2n} (\delta_0+2\delta_{{1 \over n}}+\cdots + 2\delta_{{n-1 \over n}} + \delta_1)$, so looking at the support of $\mu_2, \mu_3$ suggests the following:

Take $f(x) = \max({1 \over 2}-x,0)$.

$x_2 = {1 \over 4} f(0) = {1 \over 8} = {5 \over 40}.$

$x_3 = {1 \over 6} f(0) + {1 \over 3} f({1 \over 3}) = {1 \over 12} + {1 \over 3} ({1 \over 2}-{1 \over 3}) = {5 \over 36 } > x_2$.