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In n dimensions for n>=5 the arrangement of hyperspheres whose convex hull has minimal content is always a "sausage" (a set of hyperspheres arranged with centers along a ...

There are least two Bang's theorems, one concerning tetrahedra (Bang 1897), and the other with widths of convex domains (Bang 1951). The theorem of Bang (1897) states that ...

Line segment picking is the process of picking line segments at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a ...

Is it possible to cover completely the surface of a sphere with congruent, nonoverlapping arcs of great circles? Conway and Croft (1964) proved that it can be covered with ...

A circle packing is called rigid (or "stable") if every circle is fixed by its neighbors, i.e., no circle can be translated without disturbing other circles of the packing ...

Triangle picking is the process of picking triangles at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a random ...

A random polygon is a polygon generated in some random way. Kendall conjectured that the shape of a random polygon is close to a disk as the area of the polygon becomes large ...

Every convex body B in the Euclidean plane with area A can be inscribed in a triangle of area at most equal to 2A (Gross 1918, Eggleston 1957). The worst possible fit ...

Consider the distribution of distances l between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated ...

Cube point picking is the three-dimensional case of hypercube point picking. The average distance from a point picked at random inside a unit cube to the center is given by ...