The mean volume of boxes and cylinders circumscribed about a convex body

The mean volume of boxes circumscribed about a convex bodyK of given volume is a minimum whenK is a ball. This follows from a more general inequality, where the volume of circumscribed boxes is replaced by the product of quermassintegrals of the projections ofK on appropriate lower dimensional subspaces.     

The mean of circumscribed polygons

The mean of circumscribed polygons of a convex body in \Bbb R ²\Bbb R ^2 is again a convex body. The corresponding mapping of convex bodies is an endomorphism, and we characterize the injective case. It is shown that this endomorphism is almost always injective.     

Parallelepipeds circumscribed about a convex body in 3-space

It is proved that each convex body K ⊂ ℝ³ of volume V(K) is contained in a parallelepiped of volume . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 79–87.     

Covering the plane with convex polygons

It is proved that for any centrally symmetric convex polygonal domainP and for any natural numberr, there exists a constantk=k(P, r) such that anyk-fold covering of the plane with translates ofP can be split intor simple coverings.     

Interpolation error estimates for mean value coordinates over convex polygons

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach π.     

On the injectivity of Wachspress and mean value mappings between convex polygons

Wachspress and mean value coordinates are two generalizations of triangular barycentric coordinates to convex polygons and have recently been used to construct mappings between polygons, with application to curve deformation and image warping. We show that Wachspress mappings between convex polygons are always injective but that mean value mappings can fail to be so in extreme cases.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

Affine images of the rhombo-dodecahedron that are circumscribed about a three-dimensional convex body

The main result of the paper is dual to the author's earlier theorem on the affine images of the cube-octahedron (the convex hull of the midpoints of edges of a cube) inscribed in a three-dimensional convex body. The rhombododecahedron is the polyhedron dual to the cube-octahedron. Let us call a convex body in Κ⊂ℝ³ exceptional if it contains a parallelogram P and is contained in a cylinder with directrix P. It is proved that any nonexceptional convex body is inscribed in an affine image of the rhombo-dodecahedron. The author does not know if the assertion is true for all three-dimensional convex bodies. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 191–195. Translated by N. Yu. Netsvetaev.     

On maximal convex lattice polygons inscribed in a plane convex set

Given a convex compact setK ? ?₂ what is the largestn such thatK contains a convex latticen-gon? We answer this question asymptotically. It turns out that the maximaln is related to the largest affine perimeter that a convex set contained inK can have. This, in turn, gives a new characterization ofK ₀, the convex set inK having maximal affine perimeter.     

Aperiodic tilings of the hyperbolic plane by convex polygons

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed. Research partially supported by GIF grant no. G-454.213.06.95 and SFB 343 Bielefeld. The work of the first author was supported in part by NSF grant DMS-9424613. The work of the second author was supported in part by a grant of the Israel Academy of Sciences, and by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany).     

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume 4

Configuration spaces of convex and embedded polygons in the plane

This paper concerns the topology of configuration spaces of linkages whose underlying graph is a single cycle. Assume that the edge lengths are such that there are no configurations in which all the edges lie along a line. The main results are that, modulo translations and rotations, each component of the space of convex configurations is homeomorphic to a closed Euclidean ball and each component of the space of embedded configurations is homeomorphic to a Euclidean space. This represents an elaboration on the topological information that follows from the convexification theorem of Connelly, Demaine, and Rote.     

Transfinite mean value interpolation over polygons

Numerical Algorithms - Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications...