Relative volume increase of $\delta$-fattening of a connected set The following question was asked very recently at Relative volume increase of δ-fattening of a compact set: Is the following inequality true for all non-empty, compact sets $A \subseteq \mathbb{R}^n$ and all real $\delta>0$: 
$$\mu(A_\delta)\le \left(1+\delta\,\frac{\lambda(\partial A)}{n\,\mu(A)}\right)^n\mu(A)\tag1,$$
where $\mu$ is the Lebesgue measure over $\mathbb{R}^n$, $A_\delta$ is the $\delta$-fattening of $A$, and 
$$\lambda(\partial A)=\liminf_{\delta\downarrow 0}\delta^{-1}(\mu(A_\delta)-\mu(A))?$$
An answer to this question was given showing that in general this is false. The OP then asked if inequality (1) can be saved by (i) assuming that the set $A$ is connected and/or (ii) replacing $\lambda(\partial A)$ by $c\lambda(\partial A)$ for some real constant $c>1$.
 A: The answer to this additional question is still negative: inequality (1) cannot be saved even by (i) assuming that the set $A$ is connected and simultaneously (ii) replacing $\lambda(\partial A)$ by $c\lambda(\partial A)$ for any real constant $c>1$. 
Indeed, let $n=3$. It is easy to see (say, by projecting a small enough part of the unit sphere $S^2$ in $\mathbb{R}^3$ on an appropriate plane) that there is a universal real constant $b>0$ such that for each real $\delta>0$ there exist $N(\delta)> b/\delta^2$ distinct points $x_1,\dots,x_{N(\delta)}$ on $S^2$ with all pairwise distances between them $>2\delta$. Let $A=A(\delta)$ be the "hedgehog", which is the union of the unit ball $B$ in $\mathbb{R}^3$ and all the segments $[x_1,2x_1],\dots,[x_{N(\delta)},2x_{N(\delta)}]$. Clearly, $A$ is compact and connected.  
Moreover, the volume $\mu(A(\delta))=\mu(B)$ and the surface area $\lambda(\partial A(\delta))=\lambda(\partial B)$ do not depend on $\delta$. So, for any real $c>1$ we have 
$$\left(1+\delta\,\frac{c\lambda(\partial A(\delta))}{n\,\mu(A(\delta))}\right)^n\mu(A(\delta))
=\left(1+\delta\,\frac{c\lambda(\partial B}{n\,\mu(B)}\right)^n\mu(B)\to\mu(B)$$ 
as $\delta\downarrow 0$. 
On the other hand, for some universal real constant $a>0$ and all small enough $\delta>0$ we have 
\begin{equation*}
 \mu(A(\delta)_\delta)\ge\mu(B)+aN(\delta)\delta^2\ge\mu(B)+ab>\mu(B), 
\end{equation*}
so that 
$$\mu(A_\delta)>\left(1+\delta\,\frac{c\lambda(\partial A)}{n\,\mu(A)}\right)^n\mu(A)$$
for $A=A(\delta)$ and all small enough $\delta>0$. 
Clearly, a similar "hedgehog" construction can be done for any $n\ge2$. 
