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Diameters


bonanova
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The diameter of a closed, topologically bounded region of the plane is the greatest distance between two points in the region. Example: the diameter of a rectangle is the length of its diagonal. Of all the regions whose diameter equals 1, one of them, call it Rmax,  encloses the largest area.

Can you prove, or disprove, that Rmax also encloses all other regions of diameter 1? That is, that all other regions of diameter 1 can be made to fit inside Rmax?

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Here goes...

Rmax must be a circle of diameter one. To prove this, assume it isn’t. Then all other possible diameters of Rmax must be one or shorter than one. If this is true, then Rmax can fit inside a circle of diameter one, which has greater area than Rmax. Therefore Rmax doesn’t enclose the maximum area. By contradiction then, Rmax must be a circle of diameter one.

Not my cleanest proof, but I’m thrilled to be the first response.

Edited by CynPyn
Had hoped to hide my answer... don’t know how
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  On 9/20/2019 at 5:27 AM, CynPyn said:

Here goes...

Rmax must be a circle of diameter one. To prove this, assume it isn’t. Then all other possible diameters of Rmax must be one or shorter than one. If this is true, then Rmax can fit inside a circle of diameter one, which has greater area than Rmax. Therefore Rmax doesn’t enclose the maximum area. By contradiction then, Rmax must be a circle of diameter one.

Not my cleanest proof, but I’m thrilled to be the first response.

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Hi @CynPyn and welcome to the Den.

Let's accept from this that  Rmax is the unit diameter circle.

Now imagine a rectangle with unit diameter (diagonal has length 1.)
That can be made to fit into
Rmax .

The question is does every unit-diameter region into Rmax ?

 

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