The Pompeiu problem
نویسنده
چکیده
Let f ∈ Lloc(R) ∩ S ′, S ′ is the Schwartz class of distributions, and ∫ σ(D) f(x)dx = 0 ∀σ ∈ G, (∗) whereD ⊂ Rn, n ≥ 2, is a bounded domain, the closure D̄ of which is C1−diffeomorphic to a closed ball. Then the complement of D̄ is connected and path connected. Here G denotes the group of all rigid motions in Rn. This group consists of all translations and rotations. It is conjectured that if f 6= 0 and (∗) holds, then D is a ball. Other two conjectures, equivalent to the above one, are formulated and discussed. Three additional conjectures are formulated. Several new short proofs are given for various results. MSC: 35J05, 31B20
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