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Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets.
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If ''f'' is a strongly-convex function with parameter ''m'', then:<ref name=":0">{{Cite web |last=Nemirovsky and Ben-Tal |date=2023 |title=Optimization III: Convex Optimization |url=http://www2.isye.gatech.edu/~nemirovs/OPTIIILN2023Spring.pdf}}</ref>{{Rp|___location=Prop.6.1.4}}
* For every real number ''r'', the [[level set]] {''x'' | ''f''(''x'') ≤ ''r''} is [[Compact space|compact]].
* The function ''f'' has a unique [[global minimum]] on ''R<sup>n</sup>''.
* The [[gradient]] <math>\nabla f</math> is [[Lipschitz continuous]] with Lipschitz constant ''m''.
== Uniformly convex functions ==
A uniformly convex function,<ref name="Zalinescu">{{cite book|title=Convex Analysis in General Vector Spaces|author=C. Zalinescu|publisher=World Scientific|year=2002|isbn=9812380671}}</ref><ref name="Bauschke">{{cite book|page=[https://archive.org/details/convexanalysismo00hhba/page/n161 144]|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces |url=https://archive.org/details/convexanalysismo00hhba|url-access=limited|author=H. Bauschke and P. L. Combettes |publisher=Springer |year=2011 |isbn=978-1-4419-9467-7}}</ref> with modulus <math>\phi</math>, is a function <math>f</math> that, for all <math>x, y</math> in the ___domain and <math>t \in [0, 1],</math> satisfies
<math display="block">f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - t(1-t) \phi(\|x-y\|)</math>
where <math>\phi</math> is a function that is non-negative and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking <math>\phi(\alpha) = \tfrac{m}{2} \alpha^2</math> we recover the definition of strong convexity.
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