Editing Level set (section)

== Examples ==

Consider the 2-dimensional Euclidean distance: <math display="block">d(x, y) = \sqrt{x^2 + y^2}</math> A level set <math>L_r(d)</math> of this function consists of those points that lie at a distance of <math>r</math> from the origin, that make a [[circle]]. For example, <math>(3, 4) \in L_5(d)</math>, because <math>d(3, 4) = 5</math>. Geometrically, this means that the point <math>(3, 4)</math> lies on the circle of radius 5 centered at the origin. More generally, a [[sphere]] in a [[metric space]] <math>(M, m)</math> with radius <math>r</math> centered at <math>x \in M</math> can be defined as the level set <math>L_r(y \mapsto m(x, y))</math>.

A second example is the plot of [[Himmelblau's function]] shown in the figure to the right. Each curve shown is a level curve of the function, and  they are spaced logarithmically: if a curve represents <math>L_x</math>, the curve directly "within" represents <math>L_{x/10}</math>, and the curve directly "outside" represents <math>L_{10x}</math>.
[[File:Himmelblau contour.svg|thumb|right|Log-spaced level curve plot of [[Himmelblau's function]]<ref>{{cite journal|last=Simionescu|first=P.A.|title=Some Advancements to Visualizing Constrained Functions and Inequalities of Two Variables|journal= Journal of Computing and Information Science in Engineering|volume=11|issue=1|year=2011|doi=10.1115/1.3570770}}</ref>]]