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In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
level set of a
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f of
n variables is a set of the form
-
that is, a set where the function takes on a given constant value
c.
When the number of variables is two, a level set is generically a curve, called a
level curve,
contour lineA contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...
, or
isoline. When
n = 3, a level set is called a
level surface (see also
isosurfaceAn isosurface is a three-dimensional analog of an isoline. It is a surface that represents points of a constant value within a volume of space; in other words, it is a level set of a continuous function whose domain is 3D-space.Isosurfaces are normally displayed using computer graphics, and are...
), and for higher values of
n the level set is a
level hypersurface.
A set of the form
-
is called a
sublevel set of
f (or, alternatively, a
lower level set or
trench of
f).
-
is called a
superlevel set of
f.
A level set is a special case of a
fiber.
Properties
- The gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of f at a point is perpendicular to the level set of f at that point.
- The sublevel sets of a convex function
In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
are convex (the converse is however not generally true).
See also
- Epigraph
In mathematics, the epigraph of a function f : Rn→R is the set of points lying on or above its graph:and the strict epigraph of the function is:The set is empty if f \equiv \infty ....
- Level set method
The level set method is a numerical technique for tracking interfaces and shapes. The advantage of the level set method is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects...
- Level set (data structures)
In computer science a level set data structure is designed to represent discretely sampled dynamic level sets functions.A common use of this form of data structure is in efficient image rendering...