arXiv · 0705.4262
We present a small 4-dimensional polyhedral realization of a 2-dimensional Z-acyclic but non-contractible simplicial complex.
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Frank H. Lutz, Günter M. Ziegler. 2007-05-29. A Small Polyhedral Z-Acyclic 2-Complex in R4. https://arxiv.org/abs/0705.4262
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Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.
In a paper in Acta Mathematica Hungarica the author proved that the Morley tetrahedron of an isosceles tetrahedron, obtained by trisecting the six dihedral angles, is again isosceles, and proposed two converse conjectures. We show that both are false. There is a nonisosceles tetrahedron $T_1$ and an isosceles, nonregular tetrahedron $T_2$ whose Morley tetrahedra are regular, and there are nonisosceles tetrahedra, even a two-parameter family of tetrahedra without any symmetry, whose Morley tetrahedra are isosceles. In $T_1$ and in $T_2$ there is a pair of opposite edges such that the other four edges are equal, and we conjecture that a regular Morley tetrahedron always forces this. We prove the conjecture for every tetrahedron with a nontrivial symmetry, and we show that, up to similarity, the regular tetrahedron, $T_1$ and $T_2$ are the only tetrahedra with this edge pattern and a regular Morley tetrahedron. The tetrahedron $T_2$ has $AB=CD=1$ and $AC=AD=BC=BD=\sqrt{(21+4\sqrt6)/45}$, while $T_1$ is given by a root of a sextic with Galois group $S_6$ and cannot be expressed by radicals. We also prove that a tetrahedron with a regular Morley tetrahedron is regular if it is orthocentric, if it is isodynamic, if its three sums of opposite edges are equal, if it has three equal edges at a vertex, if it has an equilateral face, or if none of its dihedral angles is larger than $95^\circ$; the tetrahedron $T_2$ has two dihedral angles of about $98.7^\circ$. For isosceles Morley tetrahedra we conjecture that $(AB^2-CD^2)(AC^2-BD^2)(AD^2-BC^2)\ge0$ and that $AB=CD$ forces a second pair of equal opposite edges. We also conjecture that a tetrahedron with $AC=BD$ whose Morley tetrahedron satisfies $A'B'=B'C'=C'D'=D'A'$ has a nontrivial symmetry. Some proofs are computer assisted; they use exact rational arithmetic or interval arithmetic with outward rounding.
Kuperberg conjectured in 1983 that every convex body $K$ in the plane is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|K|$, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon $P$ is contained in a quadrilateral of area at most $\frac{3}{\sqrt5}\,|P|$ whose sides are parallel to sides or diagonals of $P$. This \emph{chord conjecture} implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound $\sqrt2$ remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant $\frac{3}{\sqrt5}$.