Editing Level set (section)

==Sublevel and superlevel sets==
A set of the form

: <math> L_c^-(f) = \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) \leq c \right\} </math>

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

: <math> \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) < c \right\} </math>

Similarly

: <math> L_c^+(f) = \left\{ (x_1, \dots, x_n)  \mid  f(x_1, \dots, x_n) \geq c \right\} </math>

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

: <math> \left\{ (x_1, \dots, x_n) \mid  f(x_1, \dots, x_n) > c \right\} </math>

Sublevel sets are important in [[mathematical optimization|minimization theory]]. By [[Extreme value theorem#Extension to semi-continuous functions|Weierstrass's theorem]], the [[totally bounded set|boundness]] of some [[empty set|non-empty]] sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum. The [[convex set|convexity]] of all the sublevel sets characterizes [[quasiconvex function]]s.<ref>{{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}}</ref>