Editing Level set

{{Short description|Subset of a function's domain on which its value is equal}}
{{for|the computational technique|Level-set method}}
{{redirect|Level surface|the application to force fields|Equipotential surface|water level surfaces|Equigeopotential}}
{{Use American English|date = April 2019}}

{{multiple image
|align=right
|width=140
|image1=Level sets linear function 2d.svg
|caption1=Points at constant slices of {{math|1=''x''{{sub|2}} = ''f'' (''x''{{sub|1}})}}.
|image2=Level sets linear function 3d.svg
|caption2=Lines at constant slices of {{math|1=''x''{{sub|3}} = ''f'' (''x''{{sub|1}}, ''x''{{sub|2}})}}.
|image3=Level sets linear function 4d.svg
|caption3=Planes at constant slices of {{math|1=''x''{{sub|4}} = ''f'' (''x''{{sub|1}}, ''x''{{sub|2}}, ''x''{{sub|3}})}}.
|footer={{math|(''n'' − 1)}}-dimensional level sets for functions of the form {{math|1=''f'' (''x''{{sub|1}}, ''x''{{sub|2}}, …, ''x{{sub|n}}'') = ''a''{{sub|1}}''x''{{sub|1}} + ''a''{{sub|2}}''x''{{sub|2}} + ⋯ + ''a{{sub|n}}x{{sub|n}}''}} where {{math|''a''{{sub|1}}, ''a''{{sub|2}}, …, ''a{{sub|n}}''}} are constants, in {{math|(''n'' + 1)}}-dimensional Euclidean space, for {{math|1=''n'' = 1, 2, 3}}.}}

{{multiple image
|width=140
|align=right
|image1=Level sets non-linear function 2d.svg
|caption1=Points at constant slices of {{math|1=''x''{{sub|2}} = ''f'' (''x''{{sub|1}})}}.
|image2=Level sets non-linear function 3d.svg
|caption2=Contour curves at constant slices of {{math|1=''x''{{sub|3}} = ''f'' (''x''{{sub|1}}, ''x''{{sub|2}})}}.
|image3=Level sets non-linear function 4d.svg
|caption3=Curved surfaces at constant slices of {{math|1=''x''{{sub|4}} = ''f'' (''x''{{sub|1}}, ''x''{{sub|2}}, ''x''{{sub|3}})}}.
|footer={{math|(''n'' − 1)}}-dimensional level sets of non-linear functions {{math|''f'' (''x''{{sub|1}}, ''x''{{sub|2}}, …, ''x{{sub|n}}''}}) in {{math|(''n'' + 1)}}-dimensional Euclidean space, for {{math|1=''n'' = 1, 2, 3}}.}}

In [[mathematics]], a '''level set''' of a [[real-valued function]] {{mvar|f}}  of {{mvar|n}} [[Function of several real variables|real variables]] is a [[set (mathematics)|set]] where the function takes on a given [[constant (mathematics)|constant]] value {{mvar|c}}, that is:

: <math> L_c(f) = \left\{ (x_1, \ldots, x_n)  \mid  f(x_1, \ldots, x_n) = c \right\}~. </math>

When the number of independent variables is two, a level set is called a '''level curve''', also known as ''[[contour line]]'' or ''isoline''; so a level [[curve]] is the set of all real-valued solutions of an equation in two variables {{math|''x''{{sub|1}}}} and {{math|''x''{{sub|2}}}}.  When {{math|1=''n'' = 3}}, a level set is called a '''level surface''' (or ''[[isosurface]]''); so a level [[Surface (mathematics)|surface]] is the set of all real-valued roots of an equation in three variables {{math|''x''{{sub|1}}}}, {{math|''x''{{sub|2}}}} and {{math|''x''{{sub|3}}}}. For higher values of {{mvar|n}}, the level set is a '''level hypersurface''', the set of all real-valued roots of an equation in {{math|''n'' > 3}} variables (a [[higher-dimensional]] [[hypersurface]]).

A level set is a special case of a [[fiber (mathematics)|fiber]].

==Alternative names==

[[Image:trefoil knot level curves.png|thumb|Intersections of a [[co-ordinate]] function's level surfaces with a [[trefoil knot]].  Red curves are closest to the viewer, while yellow curves are farthest.]]

Level sets show up in many applications, often under different names.  For example, an [[implicit curve]] is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an [[implicit equation]]. Analogously, a level surface is sometimes called an implicit surface or an [[isosurface]].

The name isocontour is also used, which means a contour of equal height. In various application areas, isocontours have received specific names, which indicate often the nature of the values of the considered function, such as [[isobar (meteorology)|isobar]], [[isotherm (contour line)|isotherm]], [[Contour line#Types|isogon]], [[isochrone map|isochrone]], [[isoquant]] and [[indifference curve]].

== 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>]]

==Level sets versus the gradient==
[[Image:level grad.svg|right|thumb|Consider a function ''f'' whose graph looks like a hill. The blue curves are the level sets; the red curves follow the direction of the gradient. The cautious hiker follows the blue paths; the bold hiker follows the red paths. Note that blue and red paths always cross at right angles.]]

:'''[[Theorem]]:''' If the function {{mvar|f}}  is [[differentiable function|differentiable]], the [[gradient]] of  {{mvar|f}}  at a point is either zero, or perpendicular to the level set of  {{mvar|f}} at that point.

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest. The other one is more cautious and does not want to either climb or descend, choosing a path which stays at the same height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other.

A consequence of this theorem (and its proof) is that if {{mvar|f}} is differentiable, a level set is a [[hypersurface]] and a [[manifold]] outside the [[critical point (mathematics)|critical points]] of {{mvar|f}}. At a critical point, a level set may be reduced to a point (for example at a [[local extremum]] of {{mvar|f}} ) or may have a 
[[singular point of an algebraic variety|singularity]] such as a [[intersection theory|self-intersection point]] or a [[cusp (singularity)|cusp]].

==Sublevel and superlevel sets==
A set of the form

: <math> L_c^-(f) = \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) \leq c \right\} </math>

is called a '''sublevel set''' of ''f'' (or, alternatively, a '''lower level set''' or '''trench''' of ''f''). A '''strict sublevel''' set of ''f'' is

: <math> \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) < c \right\} </math>

Similarly

: <math> L_c^+(f) = \left\{ (x_1, \dots, x_n)  \mid  f(x_1, \dots, x_n) \geq c \right\} </math>

is called a '''superlevel set''' of ''f'' (or, alternatively, an '''upper level set''' of ''f''). And a '''strict superlevel set''' of ''f'' is

: <math> \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) > c \right\} </math>

Sublevel sets are important in [[mathematical optimization|minimization theory]]. By [[Extreme value theorem#Extension to semi-continuous functions|Weierstrass's theorem]], the [[totally bounded set|boundness]] of some [[empty set|non-empty]] sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum. The [[convex set|convexity]] of all the sublevel sets characterizes [[quasiconvex function]]s.<ref>{{cite journal|last=Kiwiel|first=Krzysztof C.|title=Convergence and efficiency of subgradient methods for quasiconvex minimization|journal=Mathematical Programming, Series A|publisher=Springer|location=Berlin, Heidelberg|issn=0025-5610|pages=1–25|volume=90|issue=1|doi=10.1007/PL00011414|year=2001|mr=1819784|s2cid=10043417}}</ref>

== See also ==
* [[Epigraph (mathematics)|Epigraph]]
* [[Level-set method]]
* [[Level set (data structures)]]

==References==
{{Reflist}}

[[Category:Multivariable calculus]]
[[Category:Implicit surface modeling]]