The Krein{Milman Theorem A Project in Functional Analysis Samuel Pettersson November 29, 2016 2. Extreme points 3. The Krein{Milman theorem 4. An application Outline 1. An informal example 3. The Krein{Milman theorem 4. An application Outline 1. An informal example 2. Extreme points 4. An application Outline 1. An informal example 2. Extreme points 3. The Krein{Milman theorem Outline 1. An informal example 2. Extreme points 3. The Krein{Milman theorem 4. An application Outline 1. An informal example 2. Extreme points 3. The Krein{Milman theorem 4. An application Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 kxk2≤ 1 Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 kxk2≤ 1 Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 kxk2≤ 1 kxk1≤ 1 Convex sets and their \corners" Observation Some convex sets are the convex hulls of their \corners". kxk1≤ 1 kxk2≤ 1 kxk1≤ 1 x1; x2 ≥ 0 Convex sets and their \corners" Observation Some convex sets are not the convex hulls of their \corners". Convex sets and their \corners" Observation Some convex sets are not the convex hulls of their \corners". x1; x2 ≥ 0 Convex sets and their \corners" Observation Some convex sets are not the convex hulls of their \corners". x1; x2 ≥ 0 Convex sets and their \corners" Observation Some convex sets are not the convex hulls of their \corners". x1; x2 ≥ 0 kxk1< 1 I Formalize the notion of a corner of a convex set (extreme point). I Find a sufficient condition for a convex set to be the closed convex hull of its extreme points (Krein{Milman). Objectives I Find a sufficient condition for a convex set to be the closed convex hull of its extreme points (Krein{Milman). Objectives I Formalize the notion of a corner of a convex set (extreme point). Objectives I Formalize the notion of a corner of a convex set (extreme point). I Find a sufficient condition for a convex set to be the closed convex hull of its extreme points (Krein{Milman). Outline 1. An informal example 2. Extreme points 3. The Krein{Milman theorem 4. An application I interior points are never extremal I boundary points may be extremal I boundary points (inside the set) of strictly convex sets are always extremal Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y Remark In a normed space, Definition of an extreme point I interior points are never extremal I boundary points may be extremal I boundary points (inside the set) of strictly convex sets are always extremal Remark In a normed space, Definition of an extreme point Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y I interior points are never extremal I boundary points may be extremal I boundary points (inside the set) of strictly convex sets are always extremal Definition of an extreme point Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y Remark In a normed space, I boundary points may be extremal I boundary points (inside the set) of strictly convex sets are always extremal Definition of an extreme point Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y Remark In a normed space, I interior points are never extremal I boundary points (inside the set) of strictly convex sets are always extremal Definition of an extreme point Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y Remark In a normed space, I interior points are never extremal I boundary points may be extremal Definition of an extreme point Definition An extreme point of a convex set K ⊆ E in a vector space E is a point z 2 K not in the interior of any line segment in K: z 6= (1 − t)x + ty; 8t 2 (0; 1); 8x; y 2 K; x 6= y Remark In a normed space, I interior points are never extremal I boundary points may be extremal I boundary points (inside the set) of strictly convex sets are always extremal Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term z}|{ ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points `1 `p (1 < p < 1) Entire unit sphere `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) Entire unit sphere `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) `1 (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere (±1; ±1;::: ) Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere `1 Examples of extreme points Extreme points of the closed unit ball: Space Extreme points ith term 1 z}|{ ` ±ei = (0;:::; 0; ±1 ; 0;::: ) `p (1 < p < 1) Entire unit sphere `1 (±1; ±1;::: ) Outline 1. An informal example 2. Extreme points 3. The Krein{Milman theorem 4. An application Reminder conv A := conv A = the smallest closed and convex set containing A: Statement of Krein{Milman Theorem (Krein{Milman) A compact convex set K ⊆ E in a normed space coincides with the closed convex hull of its extreme points: K = conv(ext K) Statement of Krein{Milman Theorem (Krein{Milman) A compact convex set K ⊆ E in a normed space coincides with the closed convex hull of its extreme points: K = conv(ext K) Reminder conv A := conv A = the smallest closed and convex set containing A: I non-empty I closed I such that any line segment in K whose interior intersects M has endpoints in M: 9t 2 (0; 1): (1 − t)x + ty 2 M =) x; y 2 M; 8x; y 2 K Definition Given a compact convex set K ⊆ E in a normed space, an extreme set is a subset M ⊆ K that is Preparation for the proof: Extreme sets I non-empty I closed I such that any line segment in K whose interior intersects M has endpoints in M: 9t 2 (0; 1): (1 − t)x + ty 2 M =) x; y 2 M; 8x; y 2 K Preparation for the proof: Extreme sets Definition Given a compact convex set K ⊆ E in a normed space, an extreme set is a subset M ⊆ K that is I closed I such that any line segment in K whose interior intersects M has endpoints in M: 9t 2 (0; 1): (1 − t)x + ty 2 M =) x; y 2 M; 8x; y 2 K Preparation for the proof: Extreme sets Definition Given a compact convex set K ⊆ E in a normed space, an extreme set is a subset M ⊆ K that is I non-empty I such that any line segment in K whose interior intersects M has endpoints in M: 9t 2 (0; 1): (1 − t)x + ty 2 M =) x; y 2 M; 8x; y 2 K Preparation for the proof: Extreme sets Definition Given a compact convex set K ⊆ E in a normed space, an extreme set is a subset M ⊆ K that is I non-empty I closed Preparation for the proof: Extreme sets Definition Given a compact convex set K ⊆ E in a normed space, an extreme set is a subset M ⊆ K that is I non-empty I closed I such that any line segment in K whose interior intersects M has endpoints in M: 9t 2 (0; 1): (1 − t)x + ty 2 M =) x; y 2 M; 8x; y 2 K B := fx 2 A : hf ; xi = maxhf ; yig y2A = fmaxima of f on Ag is an extreme subset of K.