Let $f\in C^3(\mathbb R)$ with $f>0$, $$\int f(x)\:{\rm d}x=1$$ and such that $g:=(\ln f)'$ is Lipschitz continuous with Lipschitz constant $c$. Assume $$I:=\int f(x)g'(x)\:{\rm d}x>0.$$
Now let $\rho>0$, $n\in\mathbb N$ and $$\mu(x,y_1):=g(y_1)-g(x_1)-\frac\rho{d-1}\sum_{i=2}^n{g'(x_i)}^2\;\;\;\text{for }x\in\mathbb R^n\text{ and }y_1\in\mathbb R.$$ Moreover, let $$M_n:=\left\{x\in\mathbb R^n:\left|\frac1{d-1}\sum_{i=2}^n{g'(x_i)}^2-I\right|<n^{-\frac18}\right\}.$$ Are we able to show that for all sufficiently large $n$, there is a $\varepsilon>0$ with $$|\mu(x,y_1)|\ge\varepsilon\;\;\;\text{for all }x\in M_n\text{ and }y_1\in\mathbb R?\tag1$$ If not, are we at least able to show that for all fixed $y_1$, there is a $\varepsilon_{y_1}>0$ with $|\mu(x,y_1)|\ge\varepsilon_{y_1}>0$ for all $x\in M_n$?
I was able to show the following (possibly useful) properties of $f$:
- $g(y)-g(x)-g'(x)(y-x)\ge-\frac c2|y-x|^2$ for all $x,y\in\mathbb R$
- $f(y)\ge f(x)e^{\frac{|g'(x)|^2}{2c}}e^{-\frac c2\left|y-x-\frac{g'(x)}c\right|^2}\ge f(x)e^{-\frac c2\left|y-x-\frac{g'(x)}c\right|^2}$ for all $x,y\in\mathbb R$
- $f(x)\le\sqrt{\frac c{2\pi}}e^{-\frac{|g'(x)|^2}{2c}}\le\sqrt{\frac c{2\pi}}$ for all $x\in\mathbb R$
- $f$ is Lipschitz continuous with constant $\frac c{\sqrt{2\pi e}}$
Unfortunately, I wasn't able to utilize them for the desired claim.