The first point in the [tex]\tau[/tex] Manifesto is that it seems more natural to define the "Circle Constant" as

[tex]\frac{C}{r}[/tex]

instead of

[tex]\frac{C}{d},[/tex]

where [tex]C[/tex],[tex]r[/tex] and [tex]d[/tex] are respectively circle's circumference, radius and diameter. Indeed the circle is defined as the set of the points with fixed distance (radius) from a fixed center, so the radius is more "important" than the diameter.

My point is to forget that we are talking about circles, and consider a (planar) shape [tex]F[/tex], let's think of the points inside a closed simple curve. We can define its diameter as

[tex]d(F)=\sup \{ |x-y|:x,y\in F\}[/tex]

and we can also define in a standard way (e.g. by integration) its area [tex]A(F)[/tex] and perimeter [tex]P(F)[/tex]. The radius can be defined as half the diameter, but I don't see any geometrical meaning in it. If [tex]F[/tex] is convex then

[tex]P(F)\leq d(F)\pi[/tex]

i.e. the perimeter of any convex shape is always smaller than the perimeter of the circle with the same diameter. The idea behind this inequality is that for any convex set, if you draw the circle with the same diameter centered in the barycenter of the set, the the circle contains the set. (I'm not entirely sure of this, if someone has a rigorous proof he's welcome!).

Then we have the following beautiful (at least for me) variational formulation of [tex]\pi[/tex]:

[tex]\pi=\sup \{ \frac{P(F)}{d(F)}:F \text{convex}\}.[/tex]

Moreover the supremum is a maximum (e.g. the circles are minimizers, but also the so called Reuleaux polygons). So [tex]\pi[/tex] is not only the "circle constant", but is the "every convex shape maximal constant". This because the diameter is an intrinsic value of ANY shape.

I hope you agree with my argument and that it is correct. I'm not a native English speaker, so I'm sorry if there are any kind of grammatical errors. Please do not kill me for whose.