1011 problems

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.

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A Geometry

Kissing configurations in dimensions 10–31

Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.

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C Hard Geometry

The Erdős unit distance problem in the plane

Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).

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A Hard Geometry

The kissing number in dimension 5

Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.

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C Grand challenge Geometry

The Hodge conjecture

Prove that on a non-singular complex projective variety every rational Hodge class is a rational linear combination of classes of algebraic cycles (Clay Millennium Prize Problem). A full solution is not expected here.

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B Geometry · Optimization constants

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.

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B Geometry · AlphaEvolve problems

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.

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B Geometry · AlphaEvolve problems

Equidistant points in convex polygons

Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?

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B Geometry · AlphaEvolve problems

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.

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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).

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B Geometry · Optimization constants

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…

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B Geometry · Optimization constants

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.

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B Geometry · AlphaEvolve problems

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).

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B Geometry · AlphaEvolve problems

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).

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B Geometry · AlphaEvolve problems

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…

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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^∘).

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B Geometry · AlphaEvolve problems

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).

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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.

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B Geometry · AlphaEvolve problems

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.

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B Geometry · AlphaEvolve problems

Pairwise touching cylinders

Is it possible for seven infinite circular cylinders C_1,…,C_7 of unit radius to touch all the others?

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B Geometry · AlphaEvolve problems

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?

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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.

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B Geometry · AlphaEvolve problems

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.

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B Geometry · AlphaEvolve problems

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?

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B Geometry · AlphaEvolve problems

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.

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B Geometry · AlphaEvolve problems

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.

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B Geometry · AlphaEvolve problems

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?

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B Geometry · Optimization constants

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.

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B Geometry · Optimization constants

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.

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