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The Dvoretzky--Rogers finite-block estimate

Statement

Let $r \geq 2$ , let $V$ be a real or complex normed space of scalar dimension at least $r (r - 1)$ , and let $d_{1}, \dots, d_{r} > 0$ . There are $x_{1}, \dots, x_{r} \in V$ such that $∥ x_{i} ∥^{2} = d_{i}$ and, for every $A \subseteq {1, \dots, r}$ ,

${∥ \sum_{i \in A} x_{i} ∥}^{2} \leq 3 \sum_{i \in A} d_{i} .$

Facts & Assumptions

[L1]

Every nonzero finite-dimensional normed space admits a normalized biorthogonal Auerbach basis (Finite-dimensional Auerbach bases).

Proof

technique · extremal

Given: The objects and hypotheses in the Statement.

1.1

The case $r = 2$ is direct: choose any unit $u$ and put [given, L1] $x_{i} = \sqrt{d_{i}} u$ . The only nontrivial subset satisfies $∥ x_{1} + x_{2} ∥^{2} \leq 2 (d_{1} + d_{2}) < 3 (d_{1} + d_{2})$ .

L1algebra

2.1

Suppose $r \geq 3$ and put $n = r (r - 1)$ . Choose an $n$ -dimensional real [given, L1, step 1.1] subspace $W$ of $V$ (in the complex case, use the underlying real space). In Auerbach coordinates from [L1], compactness bounds the family of centered ellipsoids contained in the unit ball of $W$ , so their determinants attain a maximum. After a linear change of coordinates, take that ellipsoid to be the Euclidean unit ball.

L1algebra

3.1

Dvoretzky--Rogers' contact-point induction gives boundary points $A_{p} = (a_{p 1}, \dots, a_{p p}, 0, \dots, 0)$ , $1 \leq p \leq r$ , satisfying

givenstep 2.1

$\sum_{j < p} a_{p j}^{2} \leq \frac{p - 1}{n}, \sum_{j \leq p} a_{p j}^{2} = 1.$

For the induction step $p$ , consider the ellipsoid

$(1 + ε)^{n - p + 1} \sum_{j < p} u_{j}^{2} + (1 + ε + ε^{2})^{- (p - 1)} \sum_{j \geq p} u_{j}^{2} \leq 1.$

Its volume divided by that of the unit ball is

${(\frac{1 + ε + ε^{2}}{1 + ε})}^{(p - 1) (n - p + 1) / 2} > 1.$

Maximality therefore says that this ellipsoid is not contained in the norm unit ball. A ray to a point witnessing noncontainment meets the norm-unit boundary at a point $A (ε)$ in the interior of the ellipsoid. Since the Euclidean unit ball is contained in the norm unit ball, $∣ A (ε) ∣_{2} \geq 1$ . Letting $ε ↓ 0$ through a compact subsequence gives a common contact point $A_{p}$ and, after subtracting $∣ A (ε) ∣_{2}^{2} \geq 1$ from the ellipsoid inequality and dividing by $ε$ ,

$(n - p + 1) \sum_{j < p} a_{p j}^{2} - (p - 1) \sum_{j \geq p} a_{p j}^{2} \leq 0.$

An orthogonal rotation of the last $n - p + 1$ coordinates makes all but the $p$ -th of them zero without moving the earlier contact points. Since $∣ A_{p} ∣_{2} = 1$ , the last inequality is exactly $n \sum_{j < p} a_{p j}^{2} \leq p - 1$ . [step 2.1, maximality, compact subsequence]

4.1

The triangular form and scalar Cauchy--Schwarz now give, for real $λ_{1}, \dots, λ_{r}$ ,

givenstep 3.1

${∣ \sum_{p = 1}^{r} λ_{p} A_{p} ∣}_{2}^{2} \leq (2 + \frac{r (r - 1)}{n}) \sum_{p = 1}^{r} λ_{p}^{2} = 3 \sum_{p = 1}^{r} λ_{p}^{2} .$

Because the maximal Euclidean ball lies in the norm unit ball, the same upper bound holds for the squared norm in $W$ . [step 3.1, finite triangular sum]

5.1

Put $x_{i} = \sqrt{d_{i}} A_{i}$ and in step 4.1 take [given, step 4.1] $λ_{i} = \sqrt{d_{i}}$ for $i \in A$ and $0$ otherwise. Each $A_{i}$ lies on the norm-unit boundary, so $∥ x_{i} ∥^{2} = d_{i}$ , and the required subset inequality follows. The empty subset gives zero.

step 2.14.1∎

Depends on

Finite-dimensional Auerbach bases

Used by

Dvoretzky--Rogers theorem Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Dvoretzky and C. A. Rogers, Absolute and Unconditional Convergence in Normed Linear Spaces (standard reference, not scraped)