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  1. Jul 2018
  2. arxiv.org arxiv.org
    1. Forsimplicity, let us assume that the boundary of Ω has only one component.Letι: Σ :=∂Ω→Rnbe its isometric embedding. Letν:ι(Σ)→Sn−1be the outer unit normal. Sinceι(Σ) is assumed to be a strictly convexhypersurface inRnthere is a smooth family of embeddingsF: Σ×[0,∞]→RnwhereFt(σ) =F(σ, t) =ι(σ) +tν(ι(σ)).Note thatFt(Σ) are the ‘outer’ distance surfaces ofι(Σ). IfˆΩ denotes thebounded domain enclosed byι(Σ), then{Ft(Σ)}t≥0foliatesRn\ˆΩ and theEuclidean metric on this set can be written asG=dt2+gt,wheregtis the first fundamental form of the embeddingFt: Σ→Rn.