Level set

Examples

Consider the 2-dimensional Euclidean distance: $${\displaystyle d(x,y)=\sqrt{x^2+y^2}}$$ A level set ${\displaystyle L_r(d)}$ of this function consists of those points that lie at a distance of ${\displaystyle r}$ from the origin, that make a circle. For example, ${\displaystyle (3,4)\in L_5(d)}$, because ${\displaystyle d(3,4)=5}$. Geometrically, this means that the point ${\displaystyle (3,4)}$ lies on the circle of radius 5 centered at the origin. More generally, a sphere in a metric space ${\displaystyle (M,m)}$ with radius ${\displaystyle r}$ centered at ${\displaystyle x\in M}$ can be defined as the level set ${\displaystyle L_r(y\mapsto m(x,y))}$.

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

Theorem: If the function f is differentiable, the gradient of f 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 points of f. At a critical point, a level set may be reduced to a point (for example at a local extremum of f ) or may have a singularity such as a self-intersection point or a cusp.

Sublevel and superlevel sets

A set of the form

${\displaystyle L_c^{-}(f)=\left\{(x_1,\dots ,x_n)\mid f(x_1,\dots ,x_n)\le c\right\}}$

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

${\displaystyle \left\{(x_1,\dots ,x_n)\mid f(x_1,\dots ,x_n)<c\right\}}$

Similarly

${\displaystyle L_c^{+}(f)=\left\{(x_1,\dots ,x_n)\mid f(x_1,\dots ,x_n)\ge c\right\}}$

is called a superlevel set of f (or, alternatively, an upper level set of f). And a strict superlevel set of f is

${\displaystyle \left\{(x_1,\dots ,x_n)\mid f(x_1,\dots ,x_n)>c\right\}}$

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.