Convex equipartitions: the spicy chicken theorem

We show that, for any prime power $n$ and any convex body $K$ (i.e., a compact convex set with interior) in $\mathbb{R }^d$ , there exists a partition of $K$ into $n$ convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov–Borsuk–Ulam theorem for convex sets in the model spaces of constant curvature.     

An asymmetric affine Pólya–Szeg? principle

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya–Szeg? symmetrization principle for functions on \mathbb Rⁿ{\mathbb R^n} . Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.     

Characterization of Polytopes via Tilings with Similar Pieces

Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body K is a polytope if there are sufficiently many tilings which contain a tile similar to K. Furthermore, we give an example that this cannot be improved.     

Partitions of nonzero elements of a finite field into pairs

In this paper we prove that the nonzero elements of a finite field with odd characteristic can be partitioned into pairs with prescribed difference (maybe, with some alternatives) in each pair. The algebraic and topological approaches to such problems are considered. We also give some generalizations of these results to packing translates in a finite or infinite field, and give a short proof of a particular case of the Eliahou-Kervaire-Plaigne theorem about sum-sets.     

A Planar linear arboricity conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405–417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree Δ is either ?Δ/2? or ?(Δ + 1)/2?. In [J. L. Wu, J Graph Theory 31 (1999), 129–134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210–220], it was proven that LAC holds for all planar graphs. LAC implies that for Δ odd, la(G) = ?Δ/2?. We conjecture that for planar graphs, this equality is true also for any even Δ?6. In this article we show that it is true for any even Δ?10, leaving open only the cases Δ = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when Δ?9. © 2010 Wiley Periodicals, Inc. J Graph Theory     

Subdivision schemes for the fair discretization of the spherical motion group

We present two subdivision schemes for the fair discretization of the spherical motion group. The first one is based on the subdivision of the 600-cell according to the tetrahedral/octahedral subdivision scheme in [S. Schaefer, J. Hakenberg, J. Warren, Smooth subdivision of tetrahedral meshes, in: R. Scopigno, D. Zorin (Eds.), Eurographics Symposium on Geometry Processing, 2004, pp. 151–158]. The second presented subdivision scheme is based on the spherical kinematic mapping. In the first step we discretize an elliptic linear congruence by the icosahedral discretization of the unit sphere. Then the resulting lines of the elliptic three-space are discretized such that the difference in the maximal and minimal elliptic distance between neighboring grid points becomes minimal.     

Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences,II

We prove that the Thue–Morse sequence t along subsequences indexed by ?n ^c ? is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} ^L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) has asymptotic density 2^?L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence \(n \mapsto {t_{\left\lfloor {{n^c}} \right\rfloor }}\) attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue–Morse sequence, that is, it satisfies a Bombieri–Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} ^L is contained as an arithmetic subsequence of t.