Roundness in Analysis and Topology Stratos Prassidis Canisius College Roundness in Analysis and Topology – p. 1 Motivations and Definitions Theorem. Every Banach space is homeomorphic to a Hilbert space. Problem. Let X and Y be Banach space that are uniformly homeomorphic. Are they linearly isomorphic? Ribe: No, they are not even Lipschitz isomorphic. Johnson, Linderstauss, Schectman: Yes, if Y = ℓp, 1 < p < ∞. Unknown for large class of Banach spaces such as Lp[0, 1], ℓ1, ℓ∞. Roundness in Analysis and Topology – p. 2 Problem. When are two Banach spaces uniformly homeomorphic? Linderstauss (1963), Enflo (1968): If p, q ≥ 1 with p =6 q, then Lp(µ) is not uniformly homeomorphic to Lq(ν). Except in the case p = q′ the theorem uses roundness. Roundness in Analysis and Topology – p. 3 Definition (Enflo): A (quasi)metric space (X,d) is said to have roundness p if p is the supremum of all q that satisfy: For all {x00,x10,x01,x11} q q d(x00,x11) + d(x01,x10) ≤ q q q q d(x00,x10) + d(x00,x01) + d(x01,x11) + d(x10,x11) . Roundness in Analysis and Topology – p. 4 Per Enflo: The inequality holds for n-cubes. More n precisely, let {xε} be a collection of 2 points such that ε = (ε1,...,εn) where εi ∈{0, 1}. A pair (xε,xδ) is called an edge if the indices differ in exactly one coordinate and it is called a diagonal if the indices differ in all coordinates. Then X has roundness p iff the sum of the p-th powers of the diagonals is less than or equal to the sum of the p-th powers of the sides. Notice that there are 2n−1 diagonals and n2n−1 edges. Roundness in Analysis and Topology – p. 5 Remarks: If (X,d) is a metric space, then p ≥ 1, from the triangle inequality. If there is a pair of points of X that have a midpoint, then p ≤ 2. If the Banach space satisfies the parallelogram law, then p =2. If p < ∞, the inequality holds for q = p. Roundness in Analysis and Topology – p. 6 Results Enflo: round(Lp(µ)) = p, for 1 ≤ p ≤ 2. ′ Weston: round(Lp(µ)) = p , for 2 ≤ p ≤ ∞. Weston: round(Lp(µ)) = p, for 0 <p ≤ 1. Roundness in Analysis and Topology – p. 7 Back to the Uniform Homeomorphism Problem: Lp[0, 1] and ℓp are not uniformly homeomorphic: Enflo (1970): p =1. Bourgain (1984): 1 < p < 2. Gorelik (1994): 2 < p < ∞. Weston (1993): 0 < p < 1. Roundness in Analysis and Topology – p. 8 Tools used to prove such theorems: (1) Corson-Klee Lemma. Let f : M → Y be uniformly continuous map with M a convex subset of a Banach space and Y a metric space. The f is Lipschitz of order 1 for large distances: ∀δ > 0, ∃K(δ),d(f(x),f(y)) ≤ K(δ)kx − yk, for kx − yk≥ δ. (2) Linderstrauss, Enflo: Let f : Lp(µ) → X be a uniform homeomorphism with 1 ≤ p ≤ 2 and X a metric space of roundness q>p then f cannot satisfy a Lipschitz condition of order < (q/p) for large distances. Roundness in Analysis and Topology – p. 9 More Results Definition: A uniform Banach Group G is a Banach space X equipped with a group operation that is uniformly continuous and the zero element of X is the identity element of the group (Example: (X, +)). Problem: Are there non-commutative uniform Banach groups? Roundness in Analysis and Topology – p. 10 Remarks If φ : X → F be a uniform homeomorphism from a Banach space to a locally bounded linear space with φ(0) = 0, then x·y = φ−1(φ(x) + φ(y)) defines a uniform Banach group structure on X. For a uniform Banach group structure G on X, d(x, y) = sup kwxz − wyzk w,z∈G defines a G-invariant metric on X. Roundness in Analysis and Topology – p. 11 The intrinsic distance on X is defined as n−1 dI (x, y) = inf d(xi,xi+1) Xi=1 n where {xi}i=1 is a chain of points with x0 = x, xn = y and d(xi,xi+1) ≤ 1. The intrinsic distance is uniformly equivalent to the norm. Roundness in Analysis and Topology – p. 12 P. - Weson: Let φ : X → F be a uniform homeomorphism between a Banach space of non-trivial roundness and a quasi-normed space and G the uniform Banach group structure induced on X. Define a Banach space structure on G by: zt = φ−1(φ(z)), d (zt, 0) N(z) = lim sup I . t→∞ t Then (G, N) is uniformly homeomorphic to X and linearly isomorphic to F . In particular, F is normable with norm M(y) = N(φ−1(y)). Roundness in Analysis and Topology – p. 13 Main Points of the Proof. Since the roundness of X is p > 1, there is K > 0 such that for each z ∈ G, dI (z, 0) ≥ 1, there is u ∈ G satisfying: 2 2 1/p dI (u , z) ≤ K, |2dI (u, 0) − dI (u , 0)|≤ KdI (z , 0). Using the previous result, if di(v, 0) ≥ 1, 2n n n 1/p |dI (v , 0) − 2 dI (v, 0)|≤ 2 KdI (v, 0) Then 2n dI (v , 0) 1/p dI (v, 0) N(v) ≥ lim sup n ≥ dI (v, 0)−KdI (v, 0) ≥ . n→∞ 2 2 Roundness in Analysis and Topology – p. 14 Then Consequences (Orlicz spaces) LΦ is not uniformly homeomorphic to LΨ if there is K and p > 1 such that for s ≥ 0 and λ ≥ 1, Ψ(t) Φ(λs) ≥ K·λp′·Φ(s), lim =0. t→∞ t Lp(µ) and Lq(ν) are not uniformly homeomorphic if 0 ≤ q<p and 1 ≤ min(p, q) ≤ 2, or 0 ≤ q < 1 < p. Roundness in Analysis and Topology – p. 15 Definitions. A Banach space E has type p if there is a constant C such that: p 1/p 1/p 1 n n p ri(t)xi dt ≤ C kxik . 0 ! ! Z i=1 i=1 X X (1 ≤ p ≤ 2) . A Banach space E has cotype q if there is a constant C such that (q ≥ 2): q 1/q 1/q 1 n n q ri(t)xi dt ≥ C kxik . 0 i=1 ! i=1 ! Z Roundness in Analysis and Topology – p. 16 X X Remarks: The Rademacher functions on [0, 1] are defined as: j rj(t) = sign(sin(2 πt). (Khintchine’s Inequality) Given 0 < p < ∞, there are constants Ap and Bp such that p 1/p 2 1/2 1 n 1 n A a r (t) dt ≤ a r (t) dt p j j j j Z0 j=1 Z0 j=1 X X p 1/p 1 n ≤ Bp ajri(t) dt 0 i=1 ! Z Roundness in Analysis and Topology – p. 17 X A Banach space E is a Hilbert space iff it has type and cotype equal to 2. (Bourgain-Milman-Wolfson) A metric space (X,d) has metric type p if there is a constant B such that for all n-cubes in X: 1/2 1/2 1 − 1 l(d)2 ≤Bn p 2 l(e)2 . d,diagonalX e,Xedge (Bourgain-Milman-Wolfson) A metric space has non-trivial metric type iff it does not contain uniformly Lipschitz images of n-dimensional Hamming cubes. Roundness in Analysis and Topology – p. 18 A Banach space has type > 1 iff it has metric type > 1. Thus if a Banach space contains finite dimensional n Hamming cubes uniformly, then it contains ℓ1 ’s uniformly. For Banach space, roundness p implies metric type p (2) with B =1. Furthermore, ℓ1 ⊕∞ℓ2 has roundness 1 and type greater than 1. The main property used in the proof of the main theorem is that, for any n cube, 1/p l(dmin) ≤ B·n ·l(emax). Roundness in Analysis and Topology – p. 19 M. Mendel and A. Naor gave the appropriate definition of metric cotype and proved its properties. One sample result: If a Banach space Y coarsely embed into a Banach space X of non-trivial cotype, then qY ≤ qX . Roundness in Analysis and Topology – p. 20 Geometric Results Lafont - P. The roundness of a circle with the arc length is 1. If X is a complete metric space and the infimum of the lengths of all the homotopically non-trivial curves is realized, then X has roundness 1. A compact non-simply connected Riemannian manifold has roundness 1. Roundness in Analysis and Topology – p. 21 CAT(0)-spaces have roundness 2 (Uses comparison theorems for quadrilaterals). If the roundness of a geodesic metric space X is 2, then there is a unique geodesic joining them. Furthermore, if X is proper then X is contractible. Locally finite trees have roundness 2. Roundness is not a quasi-isometry invariant. Roundness in Analysis and Topology – p. 22 Definition For a group G, define the roundness spectrum ρ(G) = {round(ρ(Cay(G, Σ)) : Σ generates G}⊂ [1, ∞] Remarks If G is infinite finitely generated group then ρ(G) ⊂ [1, 2]. If G is finite ∞∈ ρ(G). Roundness in Analysis and Topology – p. 23 Let G be a group that contains two elements x and y such that: x2 =16 =6 y2, x =6 y±1 =6 x3, y3 =6 x±1. Then 1 ∈ ρ(G). Proof. Include x and y in a generating set Σ of G. Remove x2 and y2 if they belong to Σ. Include generators: −1 −1 −1 −1 z1 = x y, z2 = xy, z3 = xy , z4 = x y .