Level set

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In mathematics, a level set of a real-valued function f of n variables is a set of the form

$L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~,$

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 line, or isoline. When n = 3, a level set is called a level surface (see also isosurface), and for higher values of n the level set is a level hypersurface.

A set of the form

$L_c^-(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \leq c \right\}$

is called a sublevel set of f (or, alternatively, a lower level set or trench of f).

$L_c^+(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \geq c \right\}$

is called a superlevel set of f.[1][2]

A level set is a special case of a fiber.

Properties

• The gradient of f at a point is perpendicular to the level set of f at that point.
• The sublevel sets of a convex function are convex (the converse is however not generally true).