Abstract
Motivated by the elementary fact that the sum of the diagonals of a quadrilateral is less than its perimeter the following question was raised: Do the three largest sides exceed the diagonals? More generally, given n arbitrary segments in the plane, can one select $n + 1$ other segments whose endpoints are among the endpoints of the given segments and whose total length is at least as large as the total length of the given segments? Not only will the existence of these segments be shown, but also a fast $O( n\log n )$ algorithm to select them will be given. For quadrilaterals, two stronger inequalities will also be proved.
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