The Euclidean Steinitz–Bergström bound

1 · The essentials

Imagine a finite list of arrows in a space with any number of dimensions, each of length at most one. Follow the list in its given order, choosing whether to use each arrow as it is or reverse its direction. The main result says these choices can be made so that, after every step, your distance from the starting point is at most $C d$ , where $d$ is the number of dimensions. The same fixed number $C$ works regardless of how many arrows are in the list or how many dimensions there are. If the original arrows add up to zero, a related conclusion lets you rearrange them instead of reversing them, with the same distance bound after every step. The source also states that the square-root dependence on dimension cannot be improved beyond fixed numerical factors, and identifies the main manuscript as Lean-formalized.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect everyday image, imagine following a list of short walking instructions, with permission to reverse each instruction but not change its place in the list. The result guarantees a set of reversals that keeps every stopping point within a controlled distance of your start; the image is imperfect because the theorem allows any number of dimensions.

Why it matters

The bound depends on dimension rather than on the length of the sequence, so adding more vectors does not require a larger guaranteed radius. Together with the matching lower bound stated in the source, it determines the size of the Euclidean Steinitz constant up to absolute factors. The manuscript describes this as resolving the Euclidean Steinitz–Bergström conjecture.

2 · Going further

The prescribed-order problem starts with a finite sequence $v_{1}, \dots, v_{n} \in R^{d}$ satisfying $∥ v_{i} ∥_{2} \leq 1$ . The main theorem states that there are signs $ε_{i} \in {- 1, 1}$ such that $\sum_{i = 1}^{k} ε_{i} v_{i}_{2} \leq C d$ for every $1 \leq k \leq n$ . Here $C$ is an absolute constant: it does not depend on the dimension or the sequence length. One choice of signs must work for all prefixes simultaneously, and no zero-sum assumption is imposed on this signing statement.

The stated consequence concerns an indexed family satisfying $\sum_{i = 1}^{n} v_{i} = 0$ , again with $∥ v_{i} ∥_{2} \leq 1$ . There is a permutation $π$ for which $\sum_{i = 1}^{k} v_{π (i)}_{2} \leq C d$ for every prefix. Unlike the first statement, this conclusion changes the order and uses the original vectors without sign changes.

The source combines this upper bound with a matching lower bound to state $S_{2} (d) = Θ (d)$ : the Euclidean Steinitz constant has square-root growth in dimension, up to absolute multiplicative factors. It identifies this as resolving the Euclidean Steinitz–Bergström conjecture. The abstract gives the logical progression from signing to reordering to the optimal order of growth, but does not describe the construction of the signs or the argument establishing those steps. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described.

Useful notions

Vectors and vector addition
Euclidean length
Partial sums of a sequence
Signs and permutations
Bounds up to constant factors

What was known before / what is new

Before

The source identifies the Euclidean Steinitz–Bergström conjecture as the problem addressed by the manuscript. It gives no earlier quantitative bounds or historical account, so it does not support a more detailed comparison with prior results.

New

The main result supplies a single signing of any prescribed-order finite unit-ball sequence that controls every prefix by $C d$ . The manuscript states the corresponding reordering conclusion for zero-sum families. Together with the matching lower bound reported in the source, this yields the optimal order $S_{2} (d) = Θ (d)$ .

Limits and assumptions

The results concern finite sequences in Euclidean space whose vectors have length at most one. The signing theorem retains the given order; the unsigned reordering conclusion requires the family to sum to zero. The source specifies neither a numerical value of C nor an exact value of the Steinitz constant. The main result has a Lean formalization, but the supplied material does not itemize its coverage of the reordering consequence or the matching lower bound. It describes the named conjecture as resolved and lists no remaining open questions.

Mini glossary

Euclidean unit ball: The set of vectors whose Euclidean length is at most one.
Signing: A choice to multiply each vector in a sequence by either +1 or −1.
Prefix: The portion of a sequence consisting of its first k entries for some k.
Zero-sum family: A family of vectors whose total is the zero vector.
Euclidean Steinitz constant: The dimension-dependent bound needed to control all partial sums when reordering any finite zero-sum family of Euclidean unit-ball vectors.
Absolute factors: Fixed multiplicative constants that do not depend on the dimension or the sequence length.

Keywords : Euclidean Steinitz constant · Vector signing · Partial sums

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within $C d$ , independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_{2} (d) = Θ (d)$ .

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/097.md

Manuscripts in this family

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The Euclidean Steinitz–Bergström theorem

24/09/2026

Verified by Lean

Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^d has one signing for which every signed prefix has norm at most $C d$ , with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, $S_{2} (d) = Θ (d)$ , and resolves the Euclidean Steinitz–Bergström conjecture.

PDF GitHub folder Lean formalization