Sublinear function: Difference between revisions

Revision as of 23:51, 30 January 2023

Revision as of 17:05, 13 February 2023

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==Properties==

Every sublinear function is a [[convex function]]: For <math>t0 \inleq [0,t \leq 1],</math>,

:<math>

<math display=block>\begin{alignalignat}{3}

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{alignalignat}</math>

</math>

If <math>p : X \to \Reals</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}}

<math display=block>p(0) ~=~ 0 ~\leq~ p(x) + p(- x) \qquad \text{ for every } x \in X,</math>

for every <math>x \in X,</math> which implies that at least one of <math>p(x)</math> and <math>p(- x)</math> must be nonnegative; that is, for every <math>x \in X,</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}}

<math display=block>0 ~\leq~ \max \{p(x), p(- x)\} \qquad \text{ for every } x \in X.</math>

Moreover, when <math>p : X \to \Reals</math> is a sublinear function on a real vector space then the map <math>q : X \to \Reals</math> defined by <math>q(x) :~\stackrel{\scriptscriptstyle\text{def}}{=}~ \max \{p(x), p(- x)\}</math> is a seminorm.{{sfn|Narici|Beckenstein|2011|pp=120-121}}

Subadditivity of <math>p : X \to \Reals</math> guarantees that for all vectors <math>x, y \in X,</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}<ref group=proof><math>p(x) = p(y + (x - y)) \leq p(y) + p(x - y),</math> which happens if and only if <math>p(x) - p(y) \leq p(x - y).</math> <math>\blacksquare</math> Substituting <math>y := -x</math> and gives <math>p(x) - p(-x) \leq p(x - (-x)) = p(x + x) \leq p(x) + p(x),</math> which implies <math>- p(- x) \leq p(x)</math> (positive homogeneity is not needed; the triangle inequality suffices). <math>\blacksquare</math></ref>

<math display=block>p(x) - p(y) ~\leq~ p(x - y),</math>

<math display=block>- p(x) ~\leq~ p(- x),</math>

so if <math>p</math> is also [[#symmetric function|symmetric]] then the [[reverse triangle inequality]] will hold for all vectors <math> x, y \in X,</math>

<math display=block>|p(x) - p(y)| ~\leq~ p(x - y).</math>

Defining <math>\ker p :~\stackrel{\scriptscriptstyle\text{def}}{=}~ p^{-1}(0),</math> then subadditivity also guarantees that for all <math>x \in X,</math> the value of <math>p</math> on the set <math>x + (\ker p \cap -\ker p) = \{x + k : p(k) = 0 = p(-k)\}</math> is constant and equal to <math>p(x).</math><ref group=proof name=ConstantOnEquivClasses>Let <math>x \in X</math> and <math>k \in p^{-1}(0) \cap (-p^{-1}(0)).</math> It remains to show that <math>p(x + k) = p(x).</math> The triangle inequality implies <math>p(x + k) \leq p(x) + p(k) = p(x) + 0 = p(x).</math> Since <math>p(-k) = 0,</math> <math>p(x) = p(x) - p(-k) \leq p(x - (-k)) = p(x + k),</math> as desired. <math>\blacksquare</math></ref>

In particular, if <math>\ker p = p^{-1}(0)</math> is a vector subspace of <math>X</math> then <math>- \ker p = \ker p</math> and the assignment <math>x + \ker p \mapsto p(x),</math> which will be denoted by <math>\hat{p},</math> is a well-defined real-valued sublinear function on the [[Quotient space (linear algebra)|quotient space]] <math>X \,/\, \ker p</math> that satisfies <math>\hat{p} ^{-1}(0) = \ker p.</math> If <math>p</math> is a seminorm then <math>\hat{p}</math> is just the usual canonical norm on the quotient space <math>X \,/\, \ker p.</math>

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| math_statement = Suppose <math>p : X \to \Reals</math> is a sublinear functional on a vector space <math>X</math> and that <math>K \subseteq X</math> is a non-empty convex subset.

If <math>x \in X</math> is a vector and <math>a, c > 0</math> are positive real numbers such that

<math display=block>p(x) + a c ~<~ \inf_{k \in K} p\left(x + a k\right)</math>

then for every positive real <math>b > 0</math> there exists some <math>\mathbf{z} \in K</math> such that

<math display=block>p\left(x + a \mathbf{z}\right) + b c ~<~ \inf_{k \in K} p\left(x + a \mathbf{z} + b k\right).</math>

}}

Adding <math>b c</math> to both sides of the hypothesis <math display=inline>p(x) + a c \,<\, \inf_{} p\left(x + a K)</math> (where <math>p(x + a K) ~\rightstackrel{\scriptscriptstyle\text{def}}{=}~ \{p(x + a k) : k \in K\}</math>) and combining that with the conclusion gives

<math display=block>p(x) + a c + b c ~<~ \inf_{} p\left(x + a K\right) + b c ~\leq~ p\left(x + a \mathbf{z}\right) + b c ~<~ \inf_{} p\left(x + a \mathbf{z} + b K\right)</math>

which yields many more inequalities, including, for instance,

<math display=block>p(x) + a c + b c ~<~ p\left(x + a \mathbf{z}\right) + b c ~<~ p\left(x + a \mathbf{z} + b \mathbf{z}\right)</math>

in which an expression on one side of a strict inequality <math>\,<\,</math> can be obtained from the other by replacing the symbol <math>c</math> with <math>\mathbf{z}</math> (or vice versa) and moving the closing parenthesis to the right (or left) of an adjacent summand (all other symbols remain fixed and unchanged).

===Associated seminorm===

If <math>p : X \to \Reals</math> is a real-valued sublinear function on a real vector space <math>X</math> (or if <math>X</math> is complex, then when it is considered as a real vector space) then the map <math>q(x) :~\stackrel{\scriptscriptstyle\text{def}}{=}~ \max \{p(x), p(- x)\}</math> defines a [[seminorm]] on the real vector space <math>X</math> called the '''seminorm associated with <math>p.</math>'''{{sfn|Narici|Beckenstein|2011|pp=120-121}}

A sublinear function <math>p</math> on a real or complex vector space is a [[#symmetric function|symmetric function]] if and only if <math>p = q</math> where <math>q(x) :~\stackrel{\scriptscriptstyle\text{def}}{=}~ \max \{p(x), p(- x)\}</math> as before.

More generally, if <math>p : X \to \Reals</math> is a real-valued sublinear function on a (real or complex) vector space <math>X</math> then

<math display=block>q(x) ~:\stackrel{\scriptscriptstyle\text{def}}{=}~ \sup_{|u|=1} p(u x) ~=~ \sup \{p(u x) : u \text{ is a unit scalar }\}</math>

will define a [[seminorm]] on <math>X</math> if this supremum is always a real number (that is, never equal to <math>\infty</math>).

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<ol>

<li><math>p</math> is a [[linear functional]].</li>

<li>for every <math>x \in X,</math> <math>p(x) + p(- x) \leq 0.</math></li>

<li>for every <math>x \in X,</math> <math>p(x) + p(- x) = 0.</math></li>

<li><math>p</math> is a minimal sublinear function.</li>

</ol>

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<li><math>p</math> is uniformly continuous on <math>X</math>;</li>

</ol>

and if <math>p</math> is positive then wethis list may addbe extended to this listinclude:

and if <math>p</math> is positive then we may add to this list:

<ol start=4>

<li><math>\{x \in X : p(x) < 1\}</math> is open in <math>X.</math></li>

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{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=192-193}}|math_statement=

If <math>U</math> is a convex open neighborhood of the origin in a TVS[[topological vector space]] <math>X</math> then the [[Minkowski functional]] of <math>U,</math> <math>p_U : X \to [0, \infty),</math> is a continuous non-negative sublinear function on <math>X</math> such that <math>U = \left\{x \in X : p_U(x) < 1 \right\};</math> if in addition <math>U</math> is a [[balanced set]] then <math>p_U</math> is a [[seminorm]] on <math>X.</math>

if in addition <math>U</math> is [[Balanced set|balanced]] then <math>p_U</math> is a [[seminorm]] on <math>X.</math>

}}

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{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=192-193}}|math_statement=

Suppose that <math>X</math> is a TVS[[topological vector space]] (not necessarily [[Locally convex topological vector space|locally convex]] or [[Hausdorff space|Hausdorff]]) over the real or complex numbers.

Then the open convex subsets of <math>X</math> are exactly those that are of the form <math display=block>z + \{x \in X : p(x) < 1\} = \{x \in X : p(x - z) < 1\}</math> for some <math>z \in X</math> and some positive continuous sublinear function <math>p</math> on <math>X.</math>

}}

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Let <math>V</math> be an open convex subset of <math>X.</math>

If <math>0 \in V</math> then let <math>z := 0</math> and otherwise let <math>z \in V</math> be arbitrary.

Let <math>p : X \to [0, \infty)</math> be the [[Minkowski functional]] of <math>V - z,</math> where <math>p</math>which is a continuous sublinear function on <math>X</math> since <math>V - z</math> is convex, [[Absorbing set|absorbing]], and open (<math>p</math> however is not necessarily a seminorm since <math>V</math> was not assumed to be [[Balanced set|balanced]]).

From the properties of Minkowski functionals, it is known that <math>V - zX = \{x \in X : p(x)- < 1\}z,</math> fromit whichfollows that

<math>V = z + \{x \in X : p(x) < 1\}</math> follows. Since <math mathdisplay=block>z + \{x \in X : p(x) < 1\} = \{x \in X : p(x - z) < 1\},.</math> this completes the proof. [[Q.E.D.|<math>\blacksquare</math>]]

It will be shown that <math>V = z + \{x \in X : p(x) < 1\},</math> which will complete the proof.

One of the known [[Minkowski functional#Properties|properties of Minkowski functionals]] guarantees <math display=inline>\{x \in X : p(x) < 1\} = (0, 1)(V - z),</math> where <math>(0, 1)(V - z) \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \{t x : 0 < t < 1, x \in V - z\} = V - z</math> since <math>V - z</math> is convex and contains the origin.

Thus <math>V - z = \{x \in X : p(x) < 1\},</math> as desired. [[Q.E.D.|<math>\blacksquare</math>]]

}}