Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
0claims
0verified
Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
All fields Mathematics 157Computer science 18Physics & astronomy 7Chemistry & materials 9Life sciences 3Earth & society 2
30 shown
Ambidextrous Moving Sofa Constant
The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.
0claims
0verified
Block Stacking Problem
Let n ≥ 1. Let C(n) be the largest displacement that the n^th block in a stack of identical rigid rectangular blocks of width 1 can be displaced horizontally over the edge of a table, with the stack remaining stable.
0claims
0verified
Equidistant points in convex polygons
Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?
0claims
0verified
Erdős squares in a square problem
For any natural n, let C(n) denote the maximum possible sum of side lengths of n squares with disjoint interiors contained inside a unit square. Obtain upper and lower bounds for C(n) that are as strong as possible.
0claims
0verified
Favard-length decay exponent
Let E⊂ ℝ^2 be a planar set. The Favard length of E is defined by Fav(E) := 1/π∫_0^π lvert Proj R_θ Ervert dθ, where Proj is orthogonal projection to the horizontal axis and R_θ is rotation by angle θ.
0claims
0verified
Flatness constant in dimension 3
A convex body K⊂ℝ^d is hollow (lattice-free) with respect to a lattice Λ if int(K)∩Λ=∅. <a href="#CS2019-hollow-def">[CS2019-hollow-def]</a> For a hollow body, the lattice width is w(K) := min_u∈ℤ^d∖0 (max_x∈ Ku· x-min_x∈ Ku· x).
0claims
0verified
Gilbert-Pollak conjecture (Steiner ratio)
C_43 is defined as the infimum of the ratio of the length of the Steiner Minimal Tree to the length of the Euclidean Minimum Spanning Tree over all finite sets of points V ⊆ ℝ^2: C_43 = inf_VL_S(V)/L_M(V), where L_S(V) and L_M(V) denote the lengths of Steiner Minimal Tree and Minimum Spanning Tree…
0claims
0verified
Hadwiger covering / illumination number in ℝ^3
C_39=H_3 is the Hadwiger covering number in dimension 3, which can also be formulated in terms of illumination of the boundary.
0claims
0verified
Heilbronn problem in a fixed bounding box
For any n ≥ 3 and any convex body K in the plane, let C(n,K) be the largest quantity such that in every configuration of n points in K, there exists a triple of points determining a triangle of area at most C(n,K) times the area of K. Establish upper and lower bounds on C(n,K).
0claims
0verified
Heilbronn problem in an arbitrary convex bounding box
For any n ≥ 3 let C(n) be the largest quantity such that in every configuration of n points in the plane, there exists a triple of points determining a triangle of area at most C(n) times the area of their convex hull. Establish upper and lower bounds on C(n).
0claims
0verified
Kakeya needle problem
Let n ≥ 2. Let C^T(n) denote the minimal area |bigcup_j=1^n T_j| of a union of triangles T_j with vertices (x_j,0), (x_j + 1/n, 0), (x_j + j/n, 1) for some real numbers x_1,…,x_n, and similarly define C^P(n) denote the minimal area |bigcup_j=1^n P_j| of a union of parallelograms P_j with vertices…
0claims
0verified
Lebesgue universal covering constant
C_13b = a is the infimal area of a convex planar set Ω that can cover a congruent copy of every convex planar set of diameter 1.
0claims
0verified
Mahler volume product constant
Let K⊂ℝ^n be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying K=-K. Its polar body is K^∘ := y∈ℝ^n: ⟨ x,y⟩ ≤ 1 for all x∈ K. The volume product of K is vp(K) := Vol_n(K) Vol_n(K^∘).
0claims
0verified
Max to min ratios
Let n,d ≥ 2. Let C(d,n) denote the largest quantity such that, given any n distinct points x_1,…,x_n in R^d, the maximum distance max_1 ≤ i < j ≤ n ‖x_i-x_j‖ between the points is at least C(d,n) times the minimum distance min_1 ≤ i < j ≤ n ‖x_i-x_j‖. Establish upper and lower bounds for C(d,n).
0claims
0verified
Moser's convex worm cover constant
C_13a is the infimal area of a convex domain Ω that can contain a rigid motion (translation + rotation; no reflections) of every planar arc (curve, or "worm") of length 1.
0claims
0verified
Moving Sofa Constant
The moving sofa constant C_41a=A is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle.
0claims
0verified
Packing in a dilate
For any n ≥ 1 and a geometric shape P (e.g. a polygon, a polytope or a sphere), let C(n, P) denote the smallest scale s such that one can place n identical copies of P with disjoint interiors inside another copy of P scaled up by a factor of s.
0claims
0verified
Pairwise touching cylinders
Is it possible for seven infinite circular cylinders C_1,…,C_7 of unit radius to touch all the others?
0claims
0verified
Points on sphere maximizing the volume
For any n ≥ 4, Let C(n) denote the maximum volume of a polyhedron with n vertices that all lie on the unit sphere S^2. What is C(n)? Which polyhedra attain the maximum volume?
0claims
0verified
Reverse Brunn-Minkowski constant
For subsets K,L⊂ℝ^n, their Minkowski sum is K+L := x+y: x∈ K, y∈ L. In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant.
0claims
0verified
Smallest dimension in which Borsuk’s conjecture fails
For a bounded set X⊂ ℝ^n, its diameter is diam(X) := sup‖x-y‖_2: x,y∈ X. Let b(X) be the smallest integer m such that X can be written as a union X = X_1 ∪ ⋯ ∪ X_m with diam(X_i) < diam(X) for all i=1,…,m.
0claims
0verified
Sphere packing density in ℝ^4
C_36=Δ_4 is the (optimal) sphere packing density in ℝ^4, i.e. the largest fraction of ℝ^4 that can be covered by congruent balls with disjoint interiors.
0claims
0verified
Spherical Designs
A spherical t-design on the d-dimensional sphere S^d ⊂ R^d+1 is a finite set of points X ⊂ S^d such that for any polynomial P of degree at most t, the average value of P over X is equal to the average value of P over the entire sphere S^d.
0claims
0verified
Tammes problem
For N ≥ 2, let C(N) denote the maximal value of the energy E(z_1,…,z_N) := min_1 ≤ i < j ≤ N ‖z_i-z_j‖ where z_1,…,z_N range over points in S^2. Establish upper and lower bounds on C(N) that are as strong as possible. What type of configurations z_1,…,z_N come close to achieving the maximal energy?
0claims
0verified
The no 5 on a sphere problem
For n a natural number, let C(n) denote the size of the largest subset of [n]^3 = 1,…,n^3 such that no 5 points lie on a sphere or a plane. Obtain upper and lower bounds for C(n) that are as strong as possible.
0claims
0verified
The Ovals Problem
Let C denote the infimal value of λ_0(γ), the least eigenvalue of the Schrödinger operator H_γ = -d^2/ds^2 + κ^2(s) associated with a simple closed convex curve γ parameterized by arclength and normalized to have length 2π, where κ(s) is the curvature.
0claims
0verified
The three-dimensional moving sofa problem with two perpendicular turns
Define C to be the largest volume of a connected bounded subset S_3 of R^3 that can continuously pass through a three-dimensional snake-shaped corridor with a unit square cross-section, consisting of two turns in the x-y and y-z planes that are far apart. What is C?
0claims
0verified
Tight alternating knot constant
C_22b = b_o is the largest constant for which one has an inequality L ≥ b_o C for all knots that admit an alternating diagram, where L is the ropelength of a knot (or link) with crossing number) C.
0claims
0verified
Tight knot constant
C_22a is the largest constant for which one has an inequality L≥ C_22aC^3/4 for all knots, where L is the ropelength of a knot (or link) with crossing number) C.
0claims
0verified
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics