Level set

For the computational technique, seeLevel-set method.

"Level surface" redirects here. For the application to force fields, seeEquipotential surface.

Points at constant slices ofx₂=f(x₁).

Lines at constant slices ofx₃=f(x₁,x₂).

Planes at constant slices ofx₄=f(x₁,x₂,x₃).

(n− 1)-dimensional level sets for functions of the formf(x₁,x₂,…,x_n) =a₁x₁+a₂x₂+ ⋯ +a_nx_nwherea₁,a₂,…,a_nare constants, in(n+ 1)-dimensional Euclidean space, forn= 1, 2, 3.

Points at constant slices ofx₂=f(x₁).

Contour curves at constant slices ofx₃=f(x₁,x₂).

Curved surfaces at constant slices ofx₄=f(x₁,x₂,x₃).

(n− 1)-dimensional level sets of non-linear functionsf(x₁,x₂,…,x_n) in(n+ 1)-dimensional Euclidean space, forn= 1, 2, 3.

Inmathematics,alevel setof areal-valued functionfofnreal variablesis asetwhere the function takes on a givenconstantvaluec,that is:

$L_{c}(f)=\left\{(x_{1},\ldots,x_{n})\mid f(x_{1},\ldots,x_{n})=c\right\}~.$

When the number of independent variables is two, a level set is called alevelcurve,also known ascontour lineorisoline;so a level curve is the set of all real-valued solutions of an equation in two variablesx₁andx₂.Whenn= 3,a level set is called alevelsurface(orisosurface); so a level surface is the set of all real-valued roots of an equation in three variablesx₁,x₂andx₃.For higher values ofn,the level set is alevelhypersurface,the set of all real-valued roots of an equation inn> 3variables.

A level set is a special case of afiber.

Alternative names

Intersections of aco-ordinatefunction's level surfaces with atrefoil 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, animplicit curveis a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by animplicit equation.Analogously, a level surface is sometimes called an implicit surface or anisosurface.

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 asisobar,isotherm,isogon,isochrone,isoquantandindifference curve.

Examples

Consider the 2-dimensional Euclidean distance:

$$d(x,y)={\sqrt {x^{2}+y^{2}}}$$

A level set$L_{r}(d)$of this function consists of those points that lie at a distance of$r$from the origin, that make acircle.For example,$(3,4)\in L_{5}(d)$,because$d(3,4)=5$.Geometrically, this means that the point$(3,4)$lies on the circle of radius 5 centered at the origin. More generally, aspherein ametric space$(M,m)$with radius$r$centered at$x\in M$can be defined as the level set$L_{r}(y\mapsto m(x,y))$.

A second example is the plot ofHimmelblau's functionshown 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$L_{x}$,the curve directly "within" represents$L_{x/10}$,and the curve directly "outside" represents$L_{10x}$.

Log-spaced level curve plot ofHimmelblau's function

Level sets versus the gradient

Consider a functionfwhose 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 functionfisdifferentiable,thegradientoffat a point is either zero, or perpendicular to the level set offat 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 iffis differentiable, a level set is ahypersurfaceand amanifoldoutside thecritical pointsoff.At a critical point, a level set may be reduced to a point (for example at alocal extremumoff) or may have a singularitysuch as aself-intersection pointor acusp.

Sublevel and superlevel sets

A set of the form

$L_{c}^{-}(f)=\left\{(x_{1},\dots,x_{n})\mid f(x_{1},\dots,x_{n})\leq c\right\}$

is called asublevel setoff(or, alternatively, alower level setortrenchoff). Astrict sublevelset offis

$\left\{(x_{1},\dots,x_{n})\mid f(x_{1},\dots,x_{n})<c\right\}$

Similarly

$L_{c}^{+}(f)=\left\{(x_{1},\dots,x_{n})\mid f(x_{1},\dots,x_{n})\geq c\right\}$

is called asuperlevel setoff(or, alternatively, anupper level setoff). And astrict superlevel setoffis

$\left\{(x_{1},\dots,x_{n})\mid f(x_{1},\dots,x_{n})>c\right\}$

Sublevel sets are important inminimization theory.ByWeierstrass's theorem,theboundnessof somenon-emptysublevel set and the lower-semicontinuity of the function implies that a function attains its minimum. Theconvexityof all the sublevel sets characterizesquasiconvex functions.

See also

• Epigraph
• Level-set method
• Level set (data structures)