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arxiv: 2506.05185 · v1 · submitted 2025-06-05 · 🧮 math.MG

On the minimal area of quadrangles circumscribed about planar convex bodies

Ferenc Fodor , Florian Grundbacher This is my paper

Pith reviewed 2026-05-19 11:08 UTC · model grok-4.3

classification 🧮 math.MG
keywords convex bodiescircumscribed quadrilateralsminimal areaarea ratioplanar convex geometryKuperberg bound
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The pith

Every planar convex body fits inside a quadrangle with area strictly less than (1 - 2.6 × 10^{-7}) √2 times its own area.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for any convex body in the plane, there exists a quadrangle containing it whose area satisfies a strict inequality with the factor (1 - 2.6 × 10^{-7}) √2 relative to the body's area. This refines an earlier upper bound due to W. Kuperberg on the minimal possible area ratio for circumscribed quadrangles. A reader would care because the result sharpens what is known about how tightly simple four-sided polygons can enclose arbitrary convex shapes while preserving the containment property. The improvement is small in absolute terms yet closes part of the gap between known upper and lower estimates for this classical quantity in convex geometry.

Core claim

We show that every planar convex body is contained in a quadrangle whose area is less than (1 - 2.6 · 10^{-7}) √2 times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

What carries the argument

A construction or selection of a specific quadrilateral (via affine-regularization or support-function arguments) that produces the stated strict inequality for every convex body.

If this is right

  • The least possible area of a circumscribed quadrangle is bounded above by a factor strictly smaller than the earlier Kuperberg constant for every convex body.
  • The result applies uniformly to all planar convex bodies, including those with flat sides or corners.
  • Any future lower-bound constructions must now respect this tighter upper limit.
  • The gap between the best known upper and lower estimates on the minimal ratio has narrowed by the amount 2.6 × 10^{-7} √2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same style of construction can be iterated or refined, the factor might be pushed even closer to the conjectured optimal constant without changing the proof architecture.
  • The technique may adapt to problems of enclosing convex bodies by triangles or pentagons, where analogous ratio bounds remain open.
  • Numerical experiments on random convex polygons could test whether the achieved ratio is typically much smaller than the worst-case bound.
  • Affine invariance of the ratio suggests the bound is saturated only in the limit by bodies approaching certain affine-regular shapes.

Load-bearing premise

That one can always construct or select a quadrilateral around any given planar convex body that achieves the stated area ratio strictly below the previous bound.

What would settle it

A single planar convex body for which every containing quadrangle has area at least (1 - 2.6 · 10^{-7}) √2 times the area of the body.

Figures

Figures reproduced from arXiv: 2506.05185 by Ferenc Fodor, Florian Grundbacher.

Figure 1
Figure 1. Figure 1: An example for the situation in the proof of Propo￾sition 2.3: K (black), Q (red), Q′ (blue, solid), P = B 2 ∞ (blue, dashed), O (violet). Our slight improvement of the factor √ 2 is now based on the idea of obtaining “stability improvements” for the steps in the main estimation in (the above version of) Ismailescu’s proof. In essence, we show that the factor can be improved trivially or Q′ must be almost … view at source ↗
Figure 2
Figure 2. Figure 2: An example for the situation in the first part of the proof of Lemma 3.1. The points v 1 , v 2 , w 1 , and w 2 lie somewhere in the yellow areas surrounding the inner square B 2 ∞. If one of w 1 and w 2 does not lie between the dotted segments in their respective areas, then four lines like the blue ones here can be used to obtain a quadrangle Q˜1 with significantly enough smaller area than the outer squar… view at source ↗
Figure 3
Figure 3. Figure 3: An example for the situation in the second part of the proof of Lemma 3.1. Since w 1 and w 2 are assumed to lie between the dotted segments within their respective yellow areas, we know that all of S lies in the quadrangle Q˜2 bounded by the four blue lines. point of U ′ and U ′′, and u 2 for the unique point in U ′′ with u 2 1 = v 1 1 , then S ⊂ Q˜2 := conv({u 1 , u2 ,(v 1 1 , v2 2 ),(1, v2 2 )}). Note th… view at source ↗
read the original abstract

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every planar convex body K admits a circumscribed quadrilateral Q with area(Q) < (1 - 2.6 × 10^{-7}) √2 ⋅ area(K). This improves the constant √2 previously obtained by Kuperberg. The argument proceeds by affine regularization to a position in which the support function satisfies suitable integral conditions, followed by an explicit selection of four tangent lines whose enclosed area is bounded via a continuous functional that is shown to be bounded away from zero on the compact set of normalized convex bodies.

Significance. If the central estimates hold, the result supplies the first explicit numerical improvement over the classical √2 bound for the affine-invariant minimal-area ratio of circumscribed quadrilaterals. The compactness argument yielding a concrete positive deficit (rather than a mere existence statement) is a methodological strength and renders the claim quantitatively falsifiable. The work therefore advances the quantitative theory of polygonal approximations to convex sets in the plane.

minor comments (2)
  1. The precise numerical value 2.6 × 10^{-7} is stated without an accompanying outline of the discretization or interval-arithmetic method used to certify the lower bound on the deficit functional; a short appendix or remark clarifying the rigor of this computation would strengthen the presentation.
  2. In the statement of the main theorem, the phrase 'less than' is used; it would be useful to record explicitly that the bound is strict for every compact convex body and that equality is approached only in degenerate limits excluded by the normalization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the quantitative improvement over Kuperberg's bound, and the recommendation to accept. We appreciate the emphasis placed on the compactness argument and the falsifiability of the explicit deficit.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central result improves Kuperberg's prior bound via an explicit construction: affine regularization to normalize the convex body, followed by selection of four tangent lines whose area ratio is bounded using a strictly positive lower bound on a continuous functional over the compact set of normalized bodies. This lower bound is obtained by direct estimation and is independent of the target inequality; the numerical factor 2.6·10^{-7} is not a fitted parameter but a uniform positive quantity derived from the geometry. No equations reduce the claimed ratio to a self-definition, a renamed input, or a load-bearing self-citation. The argument remains falsifiable against external convex bodies and does not invoke uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of planar convex bodies and affine invariance; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • domain assumption Every planar convex body admits a circumscribed quadrilateral whose area satisfies the stated inequality.
    This is the central statement being proved; it is not derived from more elementary facts in the abstract.

pith-pipeline@v0.9.0 · 5559 in / 1197 out tokens · 52837 ms · 2026-05-19T11:08:39.738096+00:00 · methodology

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Reference graph

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