I am currently given points in the 2D plane. From which I created a convex Hull. Now I need to cover this convex hull with a polygon, but the distance between the points of convex hull and the extended polygon should be at least L or simply we can say the extended polygon should be at least L distance away from every point in the convex hull. Also, we need to minimize the perimeter of the extended polygon. I came across a solution that said
$Ans = perimeter\ of\ convex\ Hull\ +\ 2*\pi*L$
Can anyone explain what this above solution means and how it was evaluated?
The problem link is this
Consider the quadrilateral $ABCD$ as an example of a convex polygon. The locus of points a fixed distance $l$ from the outside of $ABCD$ is drawn below. This shape is the furthest away a polygon can be away from the original polygon and the perimeter of this shape is an upper bound for the perimeter for any extended convex polygon.
It is obvious that lengths $AD=LE$, $DC=FG$, $CB=HJ$ and $BA=IK$, so the perimeter of this shape is the perimeter of the quadrilateral $ABCD$ together with the arcs $EF$, $GH$, $JI$ and $KL$. These arcs form a circle if added together, with perimeter $2 \pi l$, which gives the answer to your question.