The Mathematics of P vs NP by hemant pandey - HTML preview

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We will state and prove a general theorem about shortest route through the periphery of a standard convex polygon.. We start with few definitions.

Standard convex Polygon: A standard convex polygon or SCP for short is one in which all the internal angles are between 900and 1800. A peculiar property of SCP is that all diagonals are greater than the two sides forming it, or adjacent sides to it. It is easy to establish since in a right triangle hypotenuse is diagonal or greatest side and as the opposite angle grows the diagonal side dilates. So if one angle is larger than 900  then one side i.e. side opposite to the before said angle is the largest side.

Now we are in a position to state our former result.

3.2 THEOREM

For all points lying on the periphery of a SCP, the shortest route between them is through the peripheral path.

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This can be established without any trouble. Any other route other than peripheral   route will include one or more diagonals. As stated before in SCP the diagonals are larger than the forming sides. Hence if three diagonals replace three sides they would increase the net distance.

We can prove it rigorously too as follows: -

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Space for Fig. 2