5.1 Some HOWTO questions

  5.1.1 How to show that sum of two convex functions is also convex function?
  5.1.2 What is convex Hull?
  5.1.3 Is convex full same as Polytope?
  5.1.4 What is difference between polytope and polyhedron?

This in place to keep some study notes, and other items to remember while taking this hard course.

5.1.1 How to show that sum of two convex functions is also convex function?

Let \(G\left ( u\right ) =g\left ( u\right ) +f\left ( u\right ) \) where we know \(g,f\) are two convex functions. We need to show \(G\left ( u^{\lambda }\right ) \) is also convex. Then, pick point \(u^{\lambda }\in U\,\ \)therefore

But the set \(U\) is convex (it must be, these are convex functions, so their domain is convex by definition). Pick a point \(u^{\lambda }=\left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\) where \(\lambda \in \left [ 0,1\right ] \) and \(u^{1},u^{2}\in U\). Hence we can write

\begin{align*} G\left ( u^{\lambda }\right ) & =g\left ( u^{\lambda }\right ) +f\left ( u^{\lambda }\right ) \\ G\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) & =g\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) +f\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) \end{align*}

But \(g\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) \leq \left ( 1-\lambda \right ) g\left ( u^{1}\right ) +\lambda g\left ( u^{2}\right ) \) and the same for \(f\). Then the above reduces to

\begin{align*} G\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) & \leq \left ( 1-\lambda \right ) g\left ( u^{1}\right ) +\lambda g\left ( u^{2}\right ) +\left ( 1-\lambda \right ) f\left ( u^{1}\right ) +fg\left ( u^{2}\right ) \\ & =\left ( 1-\lambda \right ) \left ( g\left ( u^{1}\right ) +f\left ( u^{1}\right ) \right ) +\lambda \left ( g\left ( u^{2}\right ) +f\left ( u^{2}\right ) \right ) \end{align*}

But \(G\left ( u\right ) =g\left ( u\right ) +f\left ( u\right ) \), then the RHS above becomes

\[ G\left ( \left ( 1-\lambda \right ) u^{1}+\lambda u^{2}\right ) \leq \left ( 1-\lambda \right ) G\left ( u^{1}\right ) +\lambda G\left ( u^{2}\right ) \]

Therefore \(G\) is a convex function.

5.1.2 What is convex Hull?

Smallest set that contains all the sets inside it, such that it is also convex. (put a closed convex ”container” around everything)

5.1.3 Is convex full same as Polytope?

No. Polytope is region which has straight edges (flat sides) and also be convex. But http://mathworld.wolfram.com/Polytope.html says ”The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points” And https://en.wikipedia.org/wiki/Polytope says ”In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope”

5.1.4 What is difference between polytope and polyhedron?

https://en.wikipedia.org/wiki/Polytope says ”In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.”

Polyhedron can be convex or not. But polyhedron can be open? While polytope not. Need to check.