I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have
$$A(\Omega\_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$
where 
$$\Omega\_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$
$A$ denotes the area and $L$ the length. I tried to find a reference for this inequality, but I only found Steiner's formula which states that equality holds when $\Omega$ is convex. Can someone please give me a reference for that inequality?



Steiner's inequality reference request
In the first note to section 4.2 of


*Schneider, Rolf*, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). [ZBL0798.52001](https://zbmath.org/?q=an:0798.52001).


Rolf Schneider cites several articles with generalizations of the Steiner formula. Of these, the following two:


*Sz.-Nagy, Béla*, Über Parallelmengen nichtkonvexer ebener Bereiche, Acta Sci. Math. 20, 36-47 (1959). [ZBL0101.14701](https://zbmath.org/?q=an:0101.14701).


*Makai, E*, Steiner type inequalities in plane geometry, Period. Polytech. Elec. Engrg. 3, 345-355 (1959). [link to pdf](https://pp.bme.hu/ee/article/view/5400/4505)


prove your inequality for simply-connected domains. For non simply-connected it follows for example by cutting the domain into pieces.


Makai also cites


*Hadwiger, Hugo*, [**Die erweiterten Steinerschen Formeln für ebene und sphärische Bereiche**](http://dx.doi.org/10.1007/BF02568102), Comment. Math. Helv. 18, 59-72 (1945). [ZBL0063.01850](https://zbmath.org/?q=an:0063.01850).


(and a four-page article in Revista Hispano-Americana, of which I found no pdf, but according to MathReviews it contains the same result as the previous article of Hadwiger) as the first reference to this inequality.
However, I have not found this result there. It speaks only about the Steiner formula for inner and outer bodies for $r$ within the injectivity radius.


