摘要
Let D be a disc with radius r in the Euclidean plane R<sup>2</sup>, and let F be a Lipschitz continuous real valued function on D. Suppose A<sub>1</sub>A<sub>2</sub>A<sub>3</sub>A<sub>4</sub> is an isosceles trapezoid with lengths of edges not greater than r, and ∠A<sub>1</sub>A<sub>2</sub>A<sub>3</sub>=α≤π/2. By means of the Brouwer fixed point theorem, it is proved that if F has a Lipschitz constant λ≤min{1, tgα}, then there exist four coplanar points in the surface M = {(x, y, F(x, y))∈R<sup>3</sup>: (x, y)∈D}which span a tetragon congruent to A<sub>1</sub>A<sub>2</sub>A<sub>3</sub>A<sub>4</sub>. In addition, some further problems are discussed.
LetD be a disc with radiusr in the Euclidean plane ?2, 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))∈?3:(x, y)?} which span a tetragon congruent toA 1 A 21 A 3 A 4. In addition, some further problems are discussed.
基金
Project supported by the National Natural Science Foundation of China (Grant No. 19231201)