MATH 5375 Partial Differential Equations

1.

Consider triangle ABC shown above.

Let P be the half plane of line AB in which point C lies, let K be the half plane of line BC in which point A lies, and let L be the half plane of line CA in which point B lies. By definition, the interior domain of the triangle ABC is the intersection P ∩ K ∩ T.

Using Hilbert’s Proposition, there exists a point D lying between A and B. using  the same reasoning, there exists a point E lying between C and D. claiming that E ∈ P ∩ K ∩ T, which is the interior of triangle ABC. Indeed, point E lies in the same half plane P of line AB as point C. Indeed, the segment EC lies entire in P since D ∗ E ∗ C. The points B, D and E lie in the same half plane T of line CA. Indeed, because of A ∗ D ∗ B and D ∗ E ∗ C, segments DB and DE lie entirely in T.

Similarly, since segments AD and DE lie in the half plane K, we see that points A, D and E lie in the same half plane K of line BC.

2.

Let assume that S is a convex having the two points A and B being in S

Lemma

If S is convex, T is convex, and then S ∩ T is convex

Let  be a half plane bounded by the line  

Let A and b be two points in the plane.

For any point P in the plane such that P the point (1 – t)P + tQ is in T

Note that A and B are on the same side of the line.

Let T be an element of the line segment AB

If A = T, then T is in half plane

If B = T, then T is in half plane

Thus every point T is an element of segment AB in.

The  is a convex set

3..  

The interior of triangle is always a convex set

Proof:

Denote the triangle as , and the interior of the boundary of  as int()

From boundary of polygon is Jordan curve, it follows that the boundary of  is equal to imag of a Jordan curve, so int() is well defined.

Denote the vertices of  as A₁, A₂, A₃ for i , put j = 1mod3 +1, k = (i + 1)mod3 +1, and:

U_i = {A_i +st(A_j – A_i)+(1 – s)t(Ak – A_i): s(0..1), t

Suppose that  A_i is an integral in  it follows from definition of polygon that A_i cannot be zero or straight. Then A_i is larger than a straight angle, which is impossible by sum angles of triangle equals two right angles.

It follows that A_i is convex

From characterization of interior of triangle, it follows that

Int() = _i

From interior of convex angle is convex set, it follows for  that U_i is a convex set.

4.

Figure A and C

Figure A

The sequence of drawing will be

A-B-C-DA-E-C

Figure C

The sequence of drawing A-B-C-D-A-E-F-C-G-D-F