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Here is a list of open problems that I like. What's annoying about them is that they remain unsolved despite my (and others') best efforts.
Warning: Some of these problems are considered "poisonous," meaning you might spend a lot of time trying to solve them and gain little from it.
A partition by orthogonal lines
A partition by orthogonal lines
Let µ be an absolutely continuous probability measure in ℝ² and assume that 0<t<½. Prove that there exist two lines a and b that split ℝ² into four regions A, B, C and D (in counter-clockwise order) such that µ(A)=µ(B)=t and µ(C)=µ(D)=½-t.
Even when the region is a centrally symmetric convex set, the problem remains unsolved.
References
References
J. Arocha, J. Jerónimo-Castro, L. Montejano, E. Roldán-Pensado. On a conjecture of Grünbaum concerning partitions of convex sets. Periodica Mathematica Hungarica, 60(1):41–47, 2010.
P. V. Blagojević, A. D. Blagojević. A problem related to Bárány–Grünbaum conjecture. Filomat, 27(1):109–113, 2013.
The width of a set of points
The width of a set of points
A band of width w in ℝ² is the closed region between two parallel lines at distance w from each other. The width of a subset X of ℝ² is the smallest w such that there is a band of width w that contains X.
Assume that there is a set P of points in ℝ² such that every subset of P with exactly 3 points has width at most 1. Prove that the width of P is at most the golden ratio (1+√5)/2.
References
References
J. J. Castro. Line transversals to translates of unit discs. Discrete & Computational Geometry, 37(3):409–417, 2007.
J. Jerónimo-Castro, E. Roldán-Pensado. Line transversals to translates of a convex body. Discrete & Computational Geometry, 45(2):329–339, 2011.
J. Jerónimo-Castro. A conjecture on line transversals to five unit discs. Boletín de la Sociedad Matemática Mexicana, 25(2):385–398, 2019.
The set of vertices of a regular pentagon gives the example showing that the number (1+√5)/2 cannot be decreased.
A hashtag partition
A hashtag partition
Another one with measures, but first a definition: Two horizontal lines and two vertical lines split ℝ² into 9 regions. Colour these regions like a chessboard and let B and W be the sets of points coloured black and white, respectively.
Assume that µ1, µ2, µ3 and µ4 are absolutely continuous probability measures in ℝ². Prove that there are two horizontal lines and two vertical lines such that the corresponding sets B and W satisfy µi(B)=µi(W)=½ for each i.
In the picture the areas of 4 regions are split. For each colour, the area of the light region equals area of the dark region.
References
References
R. N. Karasev, E. Roldán-Pensado, P. Soberón. Measure partitions using hyperplanes with fixed directions. Israel Journal of Mathematics, 212(2):705–728, 2016.
Intersecting convex sets in space
Intersecting convex sets in space
Let F be a family of convex sets in ℝ³ such that every two of them intersect. Prove that there exist three lines such that every set in F is intersected by at least one of these lines.
The picture shows five flat convex sets that pairwise intersect. Here two lines are needed in order to intersect them all.
References
References
L. Martínez-Sandoval, E. Roldán-Pensado, N. Rubin. Further consequences of the colorful Helly hypothesis. Discrete & Computational Geometry, 63(4):848–866, 2020.
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