What's the upper bound of the area of a $\psi$-thickened perimeter of a curve?

I'm reading through this paper and in the part Proof of Theorem 2 (page 4, at the bottom) I stumbled upon this problem.

Basically, we have a unit square partitioned into several districts $$D_i$$ with the area denoted by $$|D_i|$$ and perimeter by $$|\partial D_i|$$. We take some constant $$\psi$$ and make a "$$\psi$$-thickened $$\partial D_i$$", which I guess means constructing some kind of a belt around the perimeter, which has a constant width of $$\psi$$ (but not constant distance from $$\delta D_i$$). They then casually claim that this belt has an area bounded from above by $$\psi\left|\partial D_{i}\right| + \pi \psi^2.$$

That makes intuitive sense when I imagine $$D_i$$ being a polygon, and I can even prove it for a $$n$$-sided polygon. The first part of the formula corresponds to rectangles with exactly $$\psi$$ height drawn above every side of the polygon and the second part is an area of the circle which is a sum of all the circle-parts above the vertices.

The problem is, it can be any general curve that doesn't cross over itself. I'm not sure how to generalize this idea to curves.

• A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary. – Jean Marie Mar 15 '19 at 18:24
• Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions. – Sh4rP EYE Mar 15 '19 at 18:30
• Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $\psi$ = initial area + $\psi \times$ perimeter +$\pi \psi^2$ that you need a convex set. – Jean Marie Mar 15 '19 at 20:37
• Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals – Jean Marie Mar 15 '19 at 20:52
• Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly. – Sh4rP EYE Mar 16 '19 at 9:26