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Dan Brumleve
Dan Brumleve
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Center the circle and give it a unit radiusat the origin. Consider the sum of all the points. If it is zero(0,0) then the statement is true for all points. Otherwise there is an anglearc of pi in which the sum has an x value less than 0. Since the condition of the convex hull enclosing 0(0,0) is equivalent to no two adjacent points being separated by more than pi, we are done (by the pigeonhole principle).

Center the circle and give it a unit radius. Consider the sum of all the points. If it is zero then the statement is true for all points. Otherwise there is an angle of pi in which the sum has an x value less than 0. Since the condition of the convex hull enclosing 0 is equivalent to no two adjacent points being separated by more than pi, we are done.

Center the circle at the origin. Consider the sum of all the points. If it is (0,0) then the statement is true for all points. Otherwise there is an arc of pi in which the sum has an x value less than 0. Since the condition of the convex hull enclosing (0,0) is equivalent to no two adjacent points being separated by more than pi, we are done (by the pigeonhole principle).

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Dan Brumleve
Dan Brumleve
  • 2.3k
  • 17
  • 28

Center the circle and give it a unit radius. Consider the sum of all the points. If it is zero then the statement is true for all points. Otherwise there is an angle of pi in which the sum has an x value less than 0. Since the condition of the convex hull enclosing 0 is equivalent to no two adjacent points being separated by more than pi, we are done.