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Sunday, January 3, 2016

UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 14

Problem:



Solution:

Part a is obvious. The given formula is the parametric line and by definition it is in the convex set C.

Part a is the basis of the hypothesis. Consider it is true for n=k, we have

k+1i=1ti(xiyi)=ki=1ti(xiyi)+tk+1(xk+1yk+1)=(ki=1ti)ki=1tiki=1ti(xiyi)+tk+1(xk+1yk+1)

The inner sum is actually a point in C because it is a convex combination.The outer sum is also a point in C because that is also a convex combination!

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