Level set#Level sets versus the gradient

Image:trefoil knot level curves.png 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.

Consider the 2-dimensional Euclidean distance: $$d(x,\; y)\; =\; \backslash sqrt\{x^2\; +\; y^2\}$$ A level set $L\_r(d)$ of this function consists of those points that lie at a distance of $r$ from the origin, that make a circle. For example, $(3,\; 4)\; \backslash in\; L\_5(d)$, because $d(3,\; 4)\; =\; 5$. Geometrically, this means that the point $(3,\; 4)$ lies on the circle of radius 5 centered at the origin. More generally, a sphere in a metric space $(M,\; m)$ with radius $r$ centered at $x\; \backslash in\; M$ can be defined as the level set $L\_r(y\; \backslash mapsto\; m(x,\; y))$.

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 $L\_x$, the curve directly "within" represents $L\_\{x/10\}$, and the curve directly "outside" represents $L\_\{10x\}$.

File:Himmelblau contour.svg{{cite journal|last=Simionescu|first=P.A.|title=Some Advancements to Visualizing Constrained Functions and Inequalities of Two Variables|journal= Journal of Computing and Information Science in Engineering|volume=11|issue=1|year=2011|doi=10.1115/1.3570770}}]]

Image:level grad.svg

:Theorem: If the function {{mvar|f}} is differentiable, the gradient of {{mvar|f}} at a point is either zero, or perpendicular to the level set of {{mvar|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 {{mvar|f}} is differentiable, a level set is a hypersurface and a manifold outside the critical points of {{mvar|f}}. At a critical point, a level set may be reduced to a point (for example at a local extremum of {{mvar|f}} ) or may have a

singularity such as a self-intersection point or a cusp.

A set of the form

: $L\_c^-(f)\; =\; \backslash left\backslash \{\; (x\_1,\; \backslash dots,\; x\_n)\; \backslash mid\; f(x\_1,\; \backslash dots,\; x\_n)\; \backslash leq\; c\; \backslash right\backslash \}$

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

: $L\_c^+(f)\; =\; \backslash left\backslash \{\; (x\_1,\; \backslash dots,\; x\_n)\; \backslash mid\; f(x\_1,\; \backslash dots,\; x\_n)\; \backslash geq\; c\; \backslash right\backslash \}$

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

: $\backslash left\backslash \{\; (x\_1,\; \backslash dots,\; x\_n)\; \backslash mid\; f(x\_1,\; \backslash dots,\; x\_n)\; >\; c\; \backslash right\backslash \}$

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.{{cite journal|last=Kiwiel|first=Krzysztof C.|title=Convergence and efficiency of subgradient methods for quasiconvex minimization|journal=Mathematical Programming, Series A|publisher=Springer|location=Berlin, Heidelberg|issn=0025-5610|pages=1–25|volume=90|issue=1|doi=10.1007/PL00011414|year=2001|mr=1819784|s2cid=10043417}}

• Epigraph
• Level-set method
• Level set (data structures)

{{Reflist}}

Category:Multivariable calculus

Category:Implicit surface modeling