1 - convexity & hyperplanes

Convex sets

Geometric Definition

Convex sets

Algebraic Definition

Convex sets

Interpretation

Convex sets

Example

Convex sets

Altenative Definition

Convex combinations

Convex combinations

Example

Theorem, V is convex

Properties of Convex Sets

Convex Functions

Geometric Definition

Convex Functions

Algebraic Definition

Convex Functions

Epigraph

Relation

Epi - Convex sets

Properties of Convex Functions

Properties of Concave Functions

Properties of concave functions are very similar to convex,

as a function f is concave if and only if -f is convex.

Strictly Convex Functions

Geometric Definition

Strictly Convex Functions

Algebraic Definition

Strictly Convex Functions

Example

Strictly Convex Functions

Proving blueprint 1

Strictly Convex Functions

Proving blueprint 2

Level set of a function =

Quasi-convex Functions

Define the map φ : R → R, piecewise function

Hypo graph of f

Balls Notation Reminder

Hyperplane

Separating Hyperplane

Alternative definition Convexity

using separating hyperplanes

Separation Theorem (duality)

Existence separating hyper-plane

Existence separating hyper-plane

Why does V need to be closed?

Existence separating hyper-plane

Why does V need to be convex?

Supporting Hyper-planes

Relation

Separating hyperplane - Supporting hyperplane

Closure of a set =

smallest closed set that contains the original set = union of the set and all of its limit points

Characterizations for Closed Sets