Mathematics Problem Archive

LetP, P′⊂ R3 be polyhedral surfaces. Suppose that the faces of P and P′ are parallel and have the same area. Does it follow then that P and P′ are con...

Is every convex polytope unfoldable?...

Does there exist a reasonably simple algorithm for detecting the edges of a convex polyhedron intrinsically?...

Does there exist a convex polyhedron with a pseudo edge graph which is not unfoldable....

Of all convex surfaces with a fixed intrinsic diameter, is the one with the greatest area a doubled disk?...

Let S ⊂ R3 be a closed surface of constant width and fixed area. How small can the volume of S be?...

LetS⊂ R3 be a closed surface of diameter d. Suppose that there exists a constant h < dso that whenever a pair of planes separated by a distance of h i...

Let $\Gamma$ be a closed curve of fixed length $L$ in $\mathbb{R}^3$. Determine the maximum possible volume of the convex hull of $\Gamma$....

Let $\Gamma$ be a closed curve of fixed length $L$ in $\mathbb{R}^3$, and let $A$ be the area of its convex hull. Prove that $A$ is maximized when $\G...

Is every compact connected minimal surface bounded by a pair of convex planar curves topologically an annulus?...

Are there any complete negatively curved surfaces embedded in the unit ball?...

Does there exist any complete negatively curved surfaces with negative Euler characteristic contained in between a pair of parallel planes in R3....

Let M be a complete noncompact convex surface in R3, with principal curvatures k1, k2, then show that inf M|k1−k2| = 0....

The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint ...

The Erdős–Oler conjecture: when $n$ is a triangular number, packing $n-1$ circles in an equilateral triangle requires a triangle of the same size as p...

The disk covering problem about finding the smallest real number $r(n)$ such that $n$ disks of radius $r(n)$ can be arranged in such a way as to cover...

The Kobon triangle problem on triangles in line arrangements...

The Kusner conjecture: at most $2d$ points can be equidistant in $L^1$ spaces...

The McMullen problem on projectively transforming sets of points into convex position...

Opaque forest problem on finding opaque sets for various planar shapes...

For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized?...

The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\mathbb{R}^{3}$...

Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming?...

What is the lowest number of faces possible for a holyhedron?...

Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as...

Is there a general expression for the minimum ropelength of an arbitrary closed knot?...

Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it?...

Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?...

Given a unit square, does there exist a point in its plane, inside or outside the square, whose distances from all four vertices are rational? Equival...

Call a finite subset $X\subset S^2$ symmetric if a finite group acts transitively on $X$ by isometries. Determine an explicit universal upper bound fo...

Let $G$ be a planar graph circle-packed in $\mathbb{R}^2$ and let $f:V(G)\to\mathbb{Z}$ change by at most one across every edge. Add from each vertex ...

Among all finite connected vertex-transitive graphs rescaled by their diameters, which one minimizes Gromov–Hausdorff distance to the round sphere $S^...

Let the Euclidean plane or hyperbolic plane have a triangulation whose triangles have diameter at most $r$. Suppose that for every pair of radius-$r$ ...

Characterize the graphs that occur as tangency, or nerve, graphs of sphere packings with disjoint interiors in $\mathbb{R}^d$....

Prove that every sphere packing in $\mathbb{R}^3$ whose tangency graph is $\mathbb{Z}^3$ has at most one accumulation point in the one-point compactif...

If $M^n$ is a closed manifold with a strictly convex projective structure, is it cubulated?...

Suppose a hyperbolic $3$-manifold $M$ admits a CR structure, not necessarily a spherical one. Can the deformation theory of convex real projective str...

If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?...

Given $R>0$ and an integer $n\geq4$, is there an $\varepsilon>0$ and a finite set of hyperbolic $n$-simplices such that every closed cone $n$-manifold...

Rivin's experimental results appear extremely regular, possibly indicating additional structure. Investigate this phenomenon....