Level set

In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: 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 x1 and x2.

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.

Level sets show up in many applications, often under different names.

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.

of this function consists of those points that lie at a distance of

lies on the circle of radius 5 centered at the origin.

A second example is the plot of Himmelblau's function shown in the figure to the right.

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.

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.

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.
Log-spaced level curve plot of Himmelblau's function [ 1 ]
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.