This question comes from my real analysis class.For any v = ( cos ⁡ θ , sin ⁡ θ ) , θ ∈ [ 0 , 2 π ), let π v ( x ) = ⟨ v , x ⟩ . Let A , B be two bounded open convex sets in R 2 and μ be the Lebesgue measure on R 1 . If for all v , μ ( π v ( A ) ) = μ ( π v ( B ) ) , can we conclude that A differs from B by a translation, or rotation, or a reflection?My idea is that since bounded open convex set is connected and π v ( x ) is continuous, we can only concern about the value on ∂ A and ∂ B. Then I try to use the arc-length parametrization of the boundary curve to establish some equations, but I don't know how to continue. Any help is greatly appreciated!

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This question comes from my real analysis class.For any   v  =  (  cos  ⁡  θ  ,  sin  ⁡  θ  )  ,  θ  ∈  [  0  ,  2  π  ), let       π          v        (  x  )  =  ⟨  v  ,  x  ⟩  . Let   A  ,  B be two bounded open convex sets in             R              2       and   μ be the Lebesgue measure on             R              1      . If for all   v  ,  μ      (          π              v              (    A    )    )    =  μ      (          π              v              (    B    )    )  , can we conclude that   A differs from   B by a translation, or rotation, or a reflection?My idea is that since bounded open convex set is connected and       π          v        (  x  ) is continuous, we can only concern about the value on   ∂  A and   ∂  B. Then I try to use the arc-length parametrization of the boundary curve to establish some equations, but I don&#039;t know how to continue. Any help is greatly appreciated!

Leonard Mahoney · Accepted Answer

No. For instance, using lengths of projections you cannot tell apart an open unit disk and (the interior of) a Reuleaux triangle of the constant width 2.

This question comes from my real analysis class. For any v = ( cos ⁡

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