The set of closed shapes you mention only has the cardinality of the continuum, not the power set of the continuum. It takes some work to prove this, but for starters note a continuous function defined on the reals is determined by its values on the rationals.
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This is very close to the kind of path integrals that come up in physics. Eg. In string theory you might integrate over all 2d surfaces with a prescribed boundary. A few complications do come up :)
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Or ray tracing. That would be the Veach-Guibas "Metropolis Light Transport" algorithm.
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I dont think I understand, but if your set allows for infinite possibilities, you could have a unit-area rectangle with w=∞ and h=1/∞. Wouldn't that obscure your perimeter average (or any question)?
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Doesn’t it go to infinity? e.g. rectangle with unit area where one axis goes to zero.
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I believe that is the definition of one "Absolute Unit"
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Lesbesgue measure?
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*Lebesgue
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I know that the "calculus of variations" is used to prove circle has max area for a perimeter, shape of curve of fastest descent, etc. But I've never solved a problem myself with those techniques myself or know if they are developed enough to handle what you want.
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But in that case they take the familiar gradient, generalizing to infinite dimensions (each point on the curve) to define an "uphill" direction to any curve's area. Then usual solve the derivative for 0 to get the max shape without having a distribution to enumerate all curves.
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