Conditions for a monotonic integral average

Question

I am looking for conditions that ensure that an integral average of a function from $\mathbb R^n$ to $\mathbb R$ is a monotonic function of the averaging set.

To be more specific, let me start with a case of $n=1$. Let $f: \mathbb R \rightarrow \mathbb R$, $a \in \mathbb R$, $A_d$ be the interval of length $d$ centered at $a$, $\mu$ stand for the Lebesgue measure on $\mathbb R$.

Question: When $ \frac{1}{\mu(A_d)}\int_{A_d} f(x)d\mu$ is an increasing function of $d$?

Answer: This is true if $f$ is convex (I could not find this anwser in the literature but can prove it).

I am looking for an extension of this result over$f: \mathbb R^n \rightarrow \mathbb R$ (now $A_d$ is the $n$-dimensional cube with an edge of length $d$ centered at $a \in \mathbb R^n$, $\mu$ is the Lebesgue measure on $\mathbb R^n$). Would convexity of $f$ be enough?

Answer 1

If f satisfies the conditions,then fixed $a$ and let

$$ g(r)= \int_{\partial{B_r(a)}}f,h(r)=r^{n-1}g(r) $$ then

$$ \frac{1}{x^n} h(x) - \frac{n}{x^{n+1}} \int_0^x h =\frac{d}{dx} \{ \frac{1}{x^n} \int_0^x h \} \ge 0 $$ $$ \to \frac{x^n}{n} g(x) \ge \int_0^x r^{n-1}g(r) \to \frac{dg}{dr} \ge 0$$so let $v$ donated normal outward vector,and by lebesgue lemma

$$\Delta f(a)=lim_{r\to0} \frac{1}{r^n}\int_{B_r(a)}\Delta f=lim_{r\to0}\frac{1}{r^n}\int_{\partial{B_r(a)}}Df\cdot v \ge 0 $$

the converse is easy.