Figure 4: The construction of the proof of Proposition 6. The continuous curve represent the boundary of K, while the dashed one represent the level set φK ≡ tj(x). The small dots are the points of An(ε). In order to find y(x) one first finds a geodesically closest point z(x) in Zn(ε) to the radial projection x ′ of x onto ∂K. The the index j(x) is determined as one for which tj(x) is a closest point of {tj}j=0,...,mn to φK(x) in the arcosine metric. Finally y(x) is the point in [0, z(x)] such that φK(y(x)) = tj(x).
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