reference request - Name for an inequality of isoperimetric type

This inequality is essentially equivalent to the Classical isoperimetric inequality. If you have a measurable body $X$ in $mathbb{R}^n$ and a ball $Bsubset mathbb R^n$ of same volume then you have the following:
$$Area(X)=lim_{epsilon to 0} frac{operatorname{Vol}(X_{epsilon})-operatorname{Vol}(X)}{epsilon}$$
$$Area(B)=lim_{epsilon to 0} frac{operatorname{Vol}(B_{epsilon})-operatorname{Vol}(B)}{epsilon}$$
Proving that $Area(X)geq Area(B)$ follows from $operatorname{Vol}(X_{epsilon})geq operatorname{Vol}(B_{epsilon})$, which is your inequality. ($n=2$)
Now this follows from the Brunn Minkowski inequality because
$$operatorname{Vol}(X_{epsilon})=left(operatorname{Vol}(X+epsilon B)^{1/n}right)^n geq left(operatorname{Vol}(X)^{1/n}+epsilon operatorname{Vol}(B)^{1/n}right)^n=operatorname{Vol}(B_{epsilon})$$