Sublinear function: Difference between revisions

Revision as of 19:42, 3 September 2022

Revision as of 04:51, 20 November 2022

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==Properties==

Every sublinear function is a [[convex function]].: For <math>t \in [0, 1]</math>,

:<math>

Every sublinear function is a [[convex function]].

\begin{align}

p(t x + (1 - t) y)

&\leq p(t x) + p((1 - t) y) & \text{subadditivity} \\

&= t p(x) + (1 - t) p(y) & \text{nonnegative homogeneity}

\end{align}

</math>

If <math>p : X \to \R</math> is a sublinear function on a vector space <math>X</math> then<ref group=proof>If <math>x \in X</math> and <math>r := 0</math> then nonnegative homogeneity implies that <math>p(0) = p(r x) = r p(x) = 0 p(x) = 0.</math> Consequently, <math>0 = p(0) = p(x + (-x)) \leq p(x) + p(-x),</math> which is only possible if <math>0 \leq \max \{p(x), p(- x)\}.</math> <math>\blacksquare</math></ref>{{sfn|Narici|Beckenstein|2011|pp=120-121}}