Let us call a circular segment a thing like this: (| : http://en.wikipedia.org/wiki/Circular_segment

I think that the best possible constant will be achieved on doubled circular segments, i.e. things like this : ()

Indeed suppose you fix the area and diameter of you domain, and try to minimise its perimeter. Then the diameter cuts your domain into two halves. If you fix the area of one half (and minimise its preimiter), then this is a classical fact that half will be a circular segment (|.

So one juts needs to take all the formulas from wiki (on the area and perimeter of circular segments) and find the minimum.

Let us call a circular segment a thing like this: (| : https://en.wikipedia.org/wiki/Circular_segment

I think that the best possible constant will be achieved on doubled circular segments, i.e. things like this : ()

Indeed suppose you fix the area and diameter of you domain, and try to minimise its perimeter. Then the diameter cuts your domain into two halves. If you fix the area of one half (and minimise its perimiter), then this is a classical fact that half will be a circular segment (|.

So one juts needs to take all the formulas from wiki (on the area and perimeter of circular segments) and find the minimum.