A glimpse into convex geometry A glimpse into convex geometry 5 ’ \ þ ° � Ï Œ Æ
A glimpse into convex geometry Two basis reference: 1. Keith Ball, An elementary introduction to modern convex geometry 2. Chuanming Zong, What is known about unit cubes Convex geometry lies at the intersection of geometry, linear algebra, functional analysis, probability, etc.
A glimpse into convex geometry Definition A subset K ⊂ R n is called convex if tx + (1 − t ) y ⊂ K , ( ∀ x , y ∈ K and 0 ≤ t ≤ 1) . Convex body: compact convex subset with nonempty interior. It is called symmetric, if x ∈ K implies − x ∈ K .
A glimpse into convex geometry Definition A subset K ⊂ R n is called convex if tx + (1 − t ) y ⊂ K , ( ∀ x , y ∈ K and 0 ≤ t ≤ 1) . Convex body: compact convex subset with nonempty interior. It is called symmetric, if x ∈ K implies − x ∈ K .
A glimpse into convex geometry Definition A subset K ⊂ R n is called convex if tx + (1 − t ) y ⊂ K , ( ∀ x , y ∈ K and 0 ≤ t ≤ 1) . Convex body: compact convex subset with nonempty interior. It is called symmetric, if x ∈ K implies − x ∈ K .
A glimpse into convex geometry Basis facts about convex bodies Supporting hyperplane: each boundary point of a convex body admits a supporting hyperplane (not necessarily unique).
A glimpse into convex geometry Basis facts about convex bodies Separation of disjoint convex bodies by hyperplane.
A glimpse into convex geometry Hahn-Banach theorem Above two statements sound intuitively true, but an instructive proof invokes the famous Hahn-Banach theorem. Theorem If g : V → R is a sublinear function, and f : U → R is a linear functional on a subspace U of V , with f ≤ g, then there exists a linear extension ˜ f of f such that ˜ f = f on U ; ˜ f ≤ g on V .
A glimpse into convex geometry A function g : V → R is sublinear if ◮ Positive homogeneity: f ( λ x ) = λ f ( x ) , ∀ λ > 0; ◮ Subadditivity: f ( x + y ) ≤ f ( x ) + f ( y ). A convex body K containing origin defines a sublinear function (Minkowski functional): g ( x ) := inf { λ > 0 : x ∈ λ K } . A symmetric convex body defines a norm on the underlying vector space, i.e., it is sublinear (triangle inequality) and || λ x || = | λ ||| x || .
A glimpse into convex geometry A function g : V → R is sublinear if ◮ Positive homogeneity: f ( λ x ) = λ f ( x ) , ∀ λ > 0; ◮ Subadditivity: f ( x + y ) ≤ f ( x ) + f ( y ). A convex body K containing origin defines a sublinear function (Minkowski functional): g ( x ) := inf { λ > 0 : x ∈ λ K } . A symmetric convex body defines a norm on the underlying vector space, i.e., it is sublinear (triangle inequality) and || λ x || = | λ ||| x || .
A glimpse into convex geometry A function g : V → R is sublinear if ◮ Positive homogeneity: f ( λ x ) = λ f ( x ) , ∀ λ > 0; ◮ Subadditivity: f ( x + y ) ≤ f ( x ) + f ( y ). A convex body K containing origin defines a sublinear function (Minkowski functional): g ( x ) := inf { λ > 0 : x ∈ λ K } . A symmetric convex body defines a norm on the underlying vector space, i.e., it is sublinear (triangle inequality) and || λ x || = | λ ||| x || .
A glimpse into convex geometry Proof of existence of supporting hyperplane Pick x 0 ∈ ∂ K , then g ( x 0 ) = 1 by definition. Let f : R x 0 → R be f ( λ x 0 ) = λ g ( x 0 ) , λ ∈ R . It is clearly linear and dominated by g , thus there is a linear extension ˜ f , and ˜ f ≤ g . ⇒ ˜ x ∈ K = ⇒ g ( x ) ≤ 1 = f ( x ) ≤ 1 . Thus, K ⊂ { ˜ f ≤ 1 } , i.e., ˜ f = 1 is a supporting hyperplane at x 0 .
A glimpse into convex geometry John’s lemma Theorem Every n-dimensional convex body K contains a unique ellipsoid E of largest volume, and E ⊂ K ⊂ nE .
A glimpse into convex geometry Distance between convex bodies How to say two convex bodies are alike? Cube and sphere seem to be two extremal examples: for symmetric convex bodies, a cube is made of fewest possible faces, a ball can be regarded as made of infinite many faces. We need to introduce a quantitative measurement for the distance between convex bodies. A general principle for introducing distance for convex bodies is that a convex body and its images under affine transformation are all regarded as one same object. In other words, the distance should be affine invariant.
A glimpse into convex geometry Distance between convex bodies How to say two convex bodies are alike? Cube and sphere seem to be two extremal examples: for symmetric convex bodies, a cube is made of fewest possible faces, a ball can be regarded as made of infinite many faces. We need to introduce a quantitative measurement for the distance between convex bodies. A general principle for introducing distance for convex bodies is that a convex body and its images under affine transformation are all regarded as one same object. In other words, the distance should be affine invariant.
A glimpse into convex geometry Distance between convex bodies How to say two convex bodies are alike? Cube and sphere seem to be two extremal examples: for symmetric convex bodies, a cube is made of fewest possible faces, a ball can be regarded as made of infinite many faces. We need to introduce a quantitative measurement for the distance between convex bodies. A general principle for introducing distance for convex bodies is that a convex body and its images under affine transformation are all regarded as one same object. In other words, the distance should be affine invariant.
A glimpse into convex geometry Affine transformation: − → y = A · − → x + − → x 0 , where A is an invertible n × n matrix. It is a combination of a linear transformation and a translation. It preserves linear structure and convexity. Distance: d ( K , L ) = inf { λ : ˜ K ⊂ ˜ L ⊂ λ ˜ K , } where ˜ K , ˜ L are linear images of K and L . For example: d ( ball , cube ) = √ n , and d ( K , L ) = 1 means K and L are identical up to affine transformation.
A glimpse into convex geometry Affine transformation: − → y = A · − → x + − → x 0 , where A is an invertible n × n matrix. It is a combination of a linear transformation and a translation. It preserves linear structure and convexity. Distance: d ( K , L ) = inf { λ : ˜ K ⊂ ˜ L ⊂ λ ˜ K , } where ˜ K , ˜ L are linear images of K and L . For example: d ( ball , cube ) = √ n , and d ( K , L ) = 1 means K and L are identical up to affine transformation.
A glimpse into convex geometry Even though a cube looks less and less like a ball as dimension grows, we will see that there do exist slice of cube which is almost round. More precisely, The cube in R n has almost spherical sections whose dimension k is roughly log n. In fact, this statement is true for all convex bodies. This is the content of Dvoretzky’s Theorem which we are aiming for. A slice of n -cube has at most n -pair of faces. If the slice dimension is low, this sounds more plausible. So the significance is that k ∼ log n .
A glimpse into convex geometry Even though a cube looks less and less like a ball as dimension grows, we will see that there do exist slice of cube which is almost round. More precisely, The cube in R n has almost spherical sections whose dimension k is roughly log n. In fact, this statement is true for all convex bodies. This is the content of Dvoretzky’s Theorem which we are aiming for. A slice of n -cube has at most n -pair of faces. If the slice dimension is low, this sounds more plausible. So the significance is that k ∼ log n .
A glimpse into convex geometry Even though a cube looks less and less like a ball as dimension grows, we will see that there do exist slice of cube which is almost round. More precisely, The cube in R n has almost spherical sections whose dimension k is roughly log n. In fact, this statement is true for all convex bodies. This is the content of Dvoretzky’s Theorem which we are aiming for. A slice of n -cube has at most n -pair of faces. If the slice dimension is low, this sounds more plausible. So the significance is that k ∼ log n .
A glimpse into convex geometry Brunn-Minkowski inequality Original form of Brunn: Theorem (Brunn) Let K be a convex body in R n , let u be a unit vector in R n , and for each r, let H r be the hyperplane x · u = r. Then the function 1 r → vol ( K ∩ H r ) n − 1 is concave on its support.
A glimpse into convex geometry 1-d illustration
A glimpse into convex geometry A novel view by Minkowski: the slice A r := K ∩ H r viewed as convex sets in R n , then for r < s < t with s = (1 − λ ) r + λ t , A s includes the Minkowski sum (1 − λ ) A r + λ A t := { (1 − λ ) x + λ y : x ∈ A r , y ∈ A t } .
A glimpse into convex geometry Brunn’s inequality says: 1 1 1 n − 1 ≥ (1 − λ ) vol ( A r ) n − 1 + λ vol ( A t ) n − 1 . vol ( A s ) Minkowski’s formulation only involves two sets, which gets rid of the cross-section of convex body. Theorem (Brunn-Minkowski) If A and B are nonempty compact subsets of R n , then 1 1 1 n ≥ (1 − λ ) vol ( A ) n + λ vol ( B ) n . vol ((1 − λ ) A + λ B )
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