Abstract
LetD be a disc with radiusr in the Euclidean plane ℝ2, and letF be a Lipschitz continuous real valued function onD. SupposeA 1 A 21 A 3 A 4 is an isosceles trapezoid with lengths of edges not greater thanr, and ∠A 1 A 21 A 3 = α≤π/2 By means of the Brouwer fixed point theorem, it is proved that ifF has a Lipschitz constant λ≤min{1, tgα}, then there exist four coplanar points in the surfaceM = {(x, y, F(x, y))∈ℝ3:(x, y)ℝ} which span a tetragon congruent toA 1 A 21 A 3 A 4. In addition, some further problems are discussed.
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Project supported by the National Natural Science Foundation of China (Grant No. 19231201).
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Mai, J. The brouwer fixed point theorem and tetragon with all vertexes in a surface. Sci. China Ser. A-Math. 42, 18–25 (1999). https://doi.org/10.1007/BF02872046
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DOI: https://doi.org/10.1007/BF02872046