the original height of the square and its length is not bigger than α, as defined in
Lemma 8.1. We will now show that for points in the vertices of such rectangles
the thesis holds and a fortiori it is possible to obtain the conclusion for the
whole square. Figure 6 illustrates the proof.
Let x - z = kα, where k ∈ R,andα>0 is small enough. Now consider the
rectangle defined by the following points (x, x), (x, x - α), (z, x) and (z, x - α) .
From Lemma 8.1 we know that
F(x, x - α) - F(x - α, x - α) ≥ F(x, x) - F(x - α, x)
Also, from A1 we know that
F(x - α, x - α) - F(z, x - α) ≥ F(x - α, x) - F (z, x)
Adding these two inequalities we obtain that
F(x, x - α) - F(z, x - α) ≥ F(x, x) - F (z, x) (24)
Repeating the procedure, consider the rectangle defined by: (x, x - α), (x, x - 2α),
(z, x - α), (z, x - 2α). From A1 we know that:
F (x, x - 2α) - F (x - α, x - 2α) ≥ F(x, x - α) - F(x - α, x - α)
and also
F (x - 2α, x - 2α) - F (z, x - 2α) ≥ F(x - 2α, x - α) - F(z, x - α).
Using Lemma 8.1 we know that
F(x - α, x - 2α) - F(x - 2α, x - 2α) ≥ F(x - α, x - α) - F (x - 2α, x - α)
Adding the three inequalities we obtain:
F (x, x - 2α) - F (z, x - 2α) ≥ F(x, x - α) - F(z, x - α)
From (24) we obtain
F (x, x - 2α) - F (z,x - 2α) ≥ F(x, x) - F (z, x)
35
More intriguing information
1. Sex differences in the structure and stability of children’s playground social networks and their overlap with friendship relations2. ENERGY-RELATED INPUT DEMAND BY CROP PRODUCERS
3. The Employment Impact of Differences in Dmand and Production
4. Modelling Transport in an Interregional General Equilibrium Model with Externalities
5. Tax systems and tax reforms in Europe: Rationale and open issue for more radical reforms
6. PROJECTED COSTS FOR SELECTED LOUISIANA VEGETABLE CROPS - 1997 SEASON
7. The name is absent
8. A novel selective 11b-hydroxysteroid dehydrogenase type 1 inhibitor prevents human adipogenesis
9. The name is absent
10. Spatial patterns in intermunicipal Danish commuting