On the set of distances between two sets over finite fields.
On the set of weighted least squares solutions of systems of convex inequalities
On the shadows and the sections of convex sets.
On the shortest curve which meets all the lines which meet a circle
On the size of approximately convex sets in normed spaces
Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
On the Sobolev distance of convex bodies.
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the -rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...
On the speed of a planar random walk avoiding its past convex hull
On the stability of the unit circle with minimal self-perimeter in normed planes
We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.
On the Stanley Ring of a Cubical Complex.
On the structure of convex sets with applications to the moduli of spherical minimal immersions.
On the structure of sets with many k-term arithmetic progressions
On the structure of support point sets.
On the structure of the quadratic Boolean problem polytope
On the Sum of a Zonotope and an Ellipsoid
On the symmetric difference metric for convex bodies.
On the thinnest 2-fold double-lattice covering with a centrally symmetric convex domain. (Über die dünnste doppelgitterförmige 2-fache Überdeckung mit einem zentralsymmetrischen konvexen Bereich.)
On the Union of Jordan Regions and Collision-Free Translational Motion Amidst Polygonal Obstacles.
On the volume method in the study of Auerbach bases of finite-dimensional normed spaces
In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...
On the volume of a certain polytope.