Concave set

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The set in green is a non-convex set, sometimes controversially called a concave set, because it contains at least one pair of points for which the connecting segment goes outside the set.

Unlike the terms convex function and concave function that are well defined, there is no standard definition for a "concave set" in conventional mathematics.

If a set is not a convex set, then it is called a non-convex set. While some authorities proscribe the use of "concave set" as a synonym for this,[1][2] others advocate it.[3] In addition, a polygon that is not a convex polygon may sometimes be called a concave polygon.[4]

References[edit]

  1. ^ Takayama, Akira (1994), Analytical Methods in Economics, University of Michigan Press, p. 54, ISBN 9780472081356, "An often seen confusion is a "concave set". Concave and convex functions designate certain classes of functions, not of sets, whereas a convex set designates a certain class of sets, and not a class of functions. A "concave set" confuses sets with functions." 
  2. ^ Corbae, Dean; Stinchcombe, Maxwell B.; Zeman, Juraj (2009), An Introduction to Mathematical Analysis for Economic Theory and Econometrics, Princeton University Press, p. 347, ISBN 9781400833085, "There is no such thing as a concave set." 
  3. ^ Weisstein, Eric W. "Concave." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Concave.htm
  4. ^ McConnell, Jeffrey J. (2006), Computer Graphics: Theory Into Practice, p. 130, ISBN 0-7637-2250-2 .