The Brouwer fixed point theorem and tetragon with all vertexes in a surface

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.

Let D be a disc with radius r in the Euclidean plane R2, and let F be a Lipschitz continuous real valued function on D. Suppose A1A2A3A4 is an isosceles trapezoid with lengths of edges not greater than r, and ∠A1A2A3=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））∈R3: （x, y）∈D}which span a tetragon congruent to A1A2A3A4. In addition, some further problems are discussed.