Level set

In mathematics, a level set of a real-valued function Template:Mvar of Template:Mvar real variables is a set where the function takes on a given constant value Template:Mvar, 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 Template:Math and Template:Math. When Template:Math, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables Template:Math, Template:Math and Template:Math. For higher values of Template:Mvar, the level set is a level hypersurface, the set of all real-valued roots of an equation in Template:Math variables (a higher-dimensional hypersurface).

A level set is a special case of a fiber.

Alternative namesEdit

File:Trefoil knot level curves.png

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

ExamplesEdit

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

Log-spaced level curve plot of Himmelblau's function<ref>Template:Cite journal</ref>

Level sets versus the gradientEdit

File:Level grad.svg

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 Template:Mvar is differentiable, the gradient of Template:Mvar at a point is either zero, or perpendicular to the level set of Template:Mvar 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 Template:Mvar is differentiable, a level set is a hypersurface and a manifold outside the critical points of Template:Mvar. At a critical point, a level set may be reduced to a point (for example at a local extremum of Template:Mvar ) or may have a singularity such as a self-intersection point or a cusp.

Sublevel and superlevel setsEdit

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

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 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.<ref>Template:Cite journal</ref>

ReferencesEdit

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