Let $p(x) = (2\pi)^{-1/2}e^{-x^2/2}$ and let $f$ be any convex function. Some calculations have led me to believe the conjecture that $$p * f(x) \geq \frac12 (f(x+\epsilon)+f(x-\epsilon))$$ for all $\epsilon$ up to some fixed universal positive value that is independent of the convex function $f$ and real number $x$. The star denotes convolution.
Some questions I have are: is it true? How large can $\epsilon$ be?
I'd be grateful for some literature on the subject as well, addressing for instance multidimensional generalizations etc.
It seems that the answer is yes and it can be deduced from the identities:
$$\frac12(f(x+\epsilon)+f(x-\epsilon)) -f(x) = \int_\Bbb R f’’(x-u)\max\{0,\epsilon-|u|\}du$$ $$p_t*f(x)-f(x) =\int_\Bbb R f’’(x-u)\bigg[ \int_0^t p_s(u)ds\bigg]du.$$ The first of these is deduced from an annoying integration by parts. In the second identity $p_t$ denotes the standard heat kernel and the identity is derived from the fact that it solves the heat equation, then using self adjointness of the second derivative and applying fubini. Note that in both identities the integral against $f’’$ may have to be interpreted in a distributional sense as $f’’$ may in general only exist as a positive measure for convex $f$.
From the identities it is clear that the first quantity above is dominated by the second quantity uniformly over all convex functions for precisely those pairs $(\epsilon,t)$ such that $$\max\{\epsilon-|u|,0\}\leq \int_0^t p_s(u)ds,\;\;\;\; \forall u\in \Bbb R.$$ In particular for a given $t$ the claim will be true for small enough (but strictly positive!) $\epsilon$ and conversely for a given $\epsilon$ it will be true for large enough $t$.
The optimal relationship is precisely $t=\epsilon^2\pi/2.$ Furthermore the proof shows that the $f$ that achieves the optimal value has to be something whose second derivative is a delta function at 0, i.e., absolute value.
A final remark is that the proof may be slightly modified to consider things like $\frac13f(x+2\epsilon) +\frac23 f(x-\epsilon)$, in which case the kernel $\max\{0,\epsilon-|u|\}$ needs to be replaced by the appropriate function which will also be a triangular shape but with different slopes on each side. Then the optimal function and relation changes.