15.4 Additional properties and exercises

E369

[191]Let \(C⊂ ℝ^ n\) be a convex set, \(f:C→ℝ\) a convex function, and \(g:ℝ→ℝ\) a convex and weakly increasing function: prove that \(f◦ g\) is convex.

E369

[192] Let \(f:[0,∞)→ℝ\) be concave, with \(f(0)=0\) and \(f\) continuous in zero.

  • Prove that \(f\) is subadditive, i.e.

    \[ f(t)+f(s)≥ f(t+s) \]

    for every \(t,s≥ 0\). If moreover \(f\) is strictly concave and \(t{\gt}0\) then

    \[ f(t)+f(s){\gt} f(t+s)~ . \]

  • Prove that, if \(∀ x, f(x)≥ 0\), then \(f\) is weakly increasing.

  • The other way around? Find an example of \(f:[0,∞)→[0,∞)\) with \(f(0)=0\), continuous, monotonic increasing and subadditive, but not concave.

Hidden solution: [UNACCESSIBLE UUID ’193’]

E369

[194] Prove Young inequality: given \(a,b{\gt}0\) and \(p,q{\gt}1\) such that \(1/p + 1/q = 1\) then

\begin{equation} ab≤ \frac{a^ p} p+\frac{b^ q} q\label{eq:dis_ Young} \end{equation}
370

with equality if and only if \(a^ p = b^ q\); prove this using concavity of the logarithm.

See also 3. Hidden solution: [UNACCESSIBLE UUID ’195’]

E369

[196] Let \(𝛼∈ (0,1)\), show that \(x^𝛼\) is \(𝛼\)-Hölder (possibly using the above results). Hidden solution: [UNACCESSIBLE UUID ’197’]

See also exercise 376.

Distance function

E370

[198] Topics:Distance function, convex sets. Prerequisites:1,1.Let \(A⊂ ℝ^ n\) be a closed nonempty set, and \(d_ A\) the distance function defined in the exercise 1, that is \(d_ A(x)=\inf _{y∈ A} |x-y|\). Prove that \(A\) is a convex set, if and only if \(d_ A\) is a convex function.

Hidden solution: [UNACCESSIBLE UUID ’199’]

E370

[19B] Topics:Distance function, convex sets. Prerequisites:1,1.

Given \(A⊂ ℝ^ n\) a closed convex set, we define the distance function \(d_ A(x)\) as in 1; let \(z∉ A\) and \(x^*\) the projection of \(z\) on \(A\) (i.e. the point of minimum distance in the definition of \(d_ A(z)\)). Having fixed \(v=(z-x^*)/|z-x^*|\), show that \(v∈∂ f(z)\); where \(∂ f\) is the subdifferential defined in 5.

Strictly convex functions and sets

E370

[19C] Let \(C⊂ ℝ^ n\) be a convex, \(f:C→ℝ\) a convex function, and \(r∈ℝ\): then \(\{ x∈ C,f(x){\lt}r\} \) and \(\{ x∈ C,f(x)≤ r\} \) are convex (possibly empty) sets.

Remark 371

[23N]The vice versa is also true: given \(A⊂ ℝ^ n\) a closed convex set, a convex function \(f:ℝ^ n→ℝ\) such that \(A=\{ x:f(x)≤ 0\} \) always exists: For example, you can use \(f=d_ A\), as seen in 2 in the previous section.

One wonders now, what if \(f\) is strictly convex?

Definition 372

[19D] A closed convex set \(A⊂ {\mathbb {R}}^ n\) is said strictly convex if, for every \(x,y∈ A\) with \(x≠ y\) and every \(t∈ (0,1)\) you have

\[ (tx+(1-t)y)∈ {{A}^\circ }\quad . \]

(Note that a strictly convex set necessarily has a non-empty interior).

Remark 373

[19F] From the exercises 4 and 5 it follows that if \(x∈ {{A}^\circ }\) or \(y∈ {{A}^\circ }\) then \( (tx+(1-t)y)∈ {{A}^\circ }\): so the definition is ”interesting” when \(x,y∈ ∂ A\).

E373

[19G] Prerequisites:4.Let \(f:{\mathbb {R}}^ n→{\mathbb {R}}\) be a strictly convex function and \(r∈{\mathbb {R}}\) then \(A=\{ x,f(x)≤ r\} \) is a closed and strictly convex (possibly empty) set. Hidden solution: [UNACCESSIBLE UUID ’19H’]

[UNACCESSIBLE UUID ’19J’]