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Level set

Points at constant slices of x₂ = f (x₁).

Points at constant slices of x₂ = f (x₁).

Inmathematics, a level set of a real-valued function fofn real variables is a set where the function takes on a given constant value c, that is:

${\displaystyle L_{c}($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 a level curve, also known as contour lineorisoline; so a level curve is the set of all real-valued solutions of an equation in two variables x₁ and x₂. When n = 3, a level set is called a level surface (orisosurface); so a level surface is the set of all real-valued roots of an equation in three variables x₁, x₂ and x₃. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n >3 variables.

A level set is a special case of a fiber.

Alternative names[edit]

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, isotherm, isogon, isochrone, isoquant and indifference curve.

Examples[edit]

Consider the 2-dimensional Euclidean distance:

${\displaystyle L_{r}($d)}

${\displaystyle r}$

${\displaystyle (3,4)\in L_{5}($d)}

${\displaystyle d(3,4)=5}$

${\displaystyle (3,4)}$

${\displaystyle (M,m)}$

${\displaystyle r}$

${\displaystyle x\in M}$

${\displaystyle L_{r}(y\mapsto m(x,y))}$

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 ${\displaystyle L_{x}}$, the curve directly "within" represents ${\displaystyle L_{x/10}}$, and the curve directly "outside" represents ${\displaystyle L_{10x}}$.

Level sets versus the gradient[edit]

Theorem: If the function fisdifferentiable, the gradientoff at a point is either zero, or perpendicular to the level set of 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 f is differentiable, a level set is a hypersurface and a manifold outside the critical pointsoff. At a critical point, a level set may be reduced to a point (for example at a local extremumoff ) or may have a singularity such as a self-intersection point or a cusp.

Sublevel and superlevel sets[edit]

A set of the form

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

is called a sublevel setoff (or, alternatively, a lower level setortrenchoff). A strict sublevel set of fis

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

Similarly

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

is called a superlevel setoff (or, alternatively, an upper level setoff). And a strict superlevel setoffis

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

Sublevel sets are important in minimization theory. By Weierstrass's theorem, the boundness of some non-empty sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum. The convexity of all the sublevel sets characterizes quasiconvex functions.^[2]

See also[edit]

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

References[edit]