Define the diameter of a shape as the greatest distance between any two of its points. What diameter 1 shape has the greatest area?
Is it the circle?
I've been looking for the biggest little polyhedron for awhile, and looked at biggest little polygon again. A paper by Henrion and Messine extended results to the 16-gon. The trend shows that the best area is chaotic, with the optimal 14-gon having a better improvement over the regular 14-gon by a factor higher than shown in the 8, 10, and 12 gons.
If the trend continues, there may be a unit thrackle that generates a 50-gon to 90-gon of diameter 1 with a total area greater than a circle of diameter 1.
It might be possible faster than that, by using Reuleaux polygon methods.
What diameter-1 shape has the greatest area? Diameter 1 polygons that won't fit in a diameter 1 circle are easy to find, but is there one with an outside area that exceeds the uncovered area?