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Math-2240 · mathematics · Geometry

Finding matching upper and lower bounds for k-sets and halving lines

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a finite planar point set, a $k$-set is a subset of $k$ points separable from the others by a line. Determine matching upper and lower bounds for the maximum number of $k$-sets, and for the number of halving lines that split the remaining points equally.

Formulation references: [S1], [S2].

Short English statement

For a finite planar point set, a $k$-set is a subset of $k$ points separable from the others by a line. Determine matching upper and lower bounds for the maximum number of $k$-sets, and for the number of halving lines that split the remaining points equally.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2240-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2240

The claimed power saving gives a sharper planar halving-line upper bound, accompanied by generic sweep bounds at all ranks. It does not supply matching lower bounds for the $k$-set counts or halving-line counts requested here.

Finding matching upper and lower bounds for k-sets and halving lines

OpenAI's claim

Planar halving-line saving: The claimed number of halving pairs in a sufficiently large even planar configuration with no three collinear is $O(n^{4/3-\varepsilon})$ for one absolute $\varepsilon>0$. The constants are existential.

Mathematical statement of the claimed result

Mathematical claim 1.
For an even-sized planar point set $P$, with no three points collinear, let $h(P)$ count pairs whose line leaves $(|P|-2)/2$ points in each open halfplane. The claimed constants $\epsilon>0,C<\infty,n_0$ are absolute and satisfy \[h(P)\le C|P|^{4/3-\epsilon}\quad(|P|\ge n_0).\] The all-ranks theorem also bounds by $CN^{4/3-\epsilon}$ the number $S_k(P)$ of switches of the $k$ points minimizing $y-sx$ as $s$ ranges from $-\infty$ to $+\infty$, for every rank $0\le k\le N$ of a generic $N$-point configuration. This power improvement does not give matching sharp bounds at every rank. Here the all-ranks genericity conditions prohibit three collinear points, repeated first coordinates, equal slopes from different point pairs, and any proper intersection of two point-pair segments lying on a third such segment. Its parameters range over $N\ge1$ and $0\le k\le N$.

Outline of the argument

Planar halving-line saving: Rotating rank sweeps track membership changes in the lower point set, including points that later return. Complete bipartite exchange graphs and convex-chain decompositions control crossings of intervals with high activity. Near saturation yields limiting rank-coordinate measures with an impossible concentration pattern; a little-oh packing improvement at one fixed scale then contracts the recurrence.

Formal verification

Family 183 describes Lean coverage for the existential halving-pair power improvement and the stated generic sweep-level estimate.

Planar halving-line saving: The manuscript links HalvingLines.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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