Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$
where $A_r$ is the $r$-enlargement of $A$ with respect to the metric $d$.
The isoperimetric profile of $\mu$ is then defined for $p \in [0, 1]$ as
$$I_\mu (p) := \inf \left\{ \mu^+ (A) : \mu(A) = p \right\}.$$
I am primarily interested in probability measures on $\mathbf{R}^d$ equipped with the standard Euclidean metric.
A basic observation is that for "nice" $\mu$, the isoperimetric profile should be invariant under the mapping $p \mapsto 1 - p$. The intuition is that for reasonable sets $A$, it will hold that $\mu^+ (A) = \mu^+ \left( A^\complement \right)$, and that even if this fails for specific $A$, it may not hold for the $A$ which is extremal in the $\inf$ which defines $I_\mu$. Still, I think that it is known not to hold in complete generality.
My issue is that most of the references which I have seen are very vague about the actual conditions under which $I_\mu$ is indeed symmetric in this way. What I would like is to identify a reference which gives a concrete statement of some appropriately general conditions on $\mu$ which ensure this.
The general thrust of these references seems to suggest that if $\mu$ has a density with respect to Lebesgue measure which is locally bounded (i.e. on each ball of finite radius), then this is sufficient for the symmetry to hold. Life should presumably be even easier if this density is e.g. continuous (and this would largely suffice for my application), but I'd prefer to get the more general result if possible; assuming smoothness on the density would certainly be a step too far.
I don't need to understand the nuts and bolts of the result itself, but if there's a reference which can explain the proof reasonably well, then this would of course be greatly appreciated.
