The spherical conjecture in Minkowski geometry

We show that the shapes of convex bodies containing m translates of a convex body K, so that their Minkowskian surface area is minimum, tends for growing m to a convex body L.Received: 7 January 2002     

Inhomogeneous Minkowski conjecture (letter to the editor)

The author corrects errors which occurred in his papers published in J. Sov. Math.,23, No. 2 (1983). A new bound of the inhomogeneous minimum Mn is given: forn?200 $$M_n \leqslant 2^{ - \tfrac{n}{2}} \left( {4.4455 + \tfrac{{1.3545}}{{n - 1}}} \right).$$ This is stronger than Davenport's result forn?200.     

Towards a proof of the 24-cell conjecture

This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the lattice packing D₄ is the densest sphere packing in 4 dimensions.     

On a conjecture of the Euler numbers

The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using the analytic methods. That is, for any prime and integer α?1 we proved , where E2n are the Euler numbers and ?(n) the Euler function.     

Applications of various inequalities to Minkowski geometry

Given a finite-dimensional normed space with unit ballB, a natural question to ask is how small (or big) can the surface areas ofB (measured in its own metric) be. Using two different definitions of surface areas we give lower bounds for this quantity. In a separate section, we also show that (using one of the definitions of surface area) a suitably normalized solution to the isoperimetric problem is equal to the unit ball if and only if the ball is an ellipsoid.     

Geometric proof of a conjecture of Fulton

We give a geometric proof of a conjecture of Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).     

A pascal theorem applied to Minkowski geometry

If on an oval in a projective plane a 4-point Pascal theorem, , with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} (O) (), with c O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of , then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.     

On Eckhoff's conjecture for Radon numbers; or how far the proof is still away

Eckhoff's conjecture for the Τ-Radon numbers r(Τ) of a convexity space. (X,C) says r(Τ) ≦ (r?1)(Τ?1)+1, with r = r(2). The main result of this note is that Eckhoff's conjecture is true in case ¦X¦ ≦ 2r and Τ = 3, i.e. each (2r?1)-set in a space with 2r?1 or 2r elements has a 3-Radon partition.     

A proof of a consequence of Dirac's conjecture

Geometriae Dedicata -     

The proof of hermann''s conjecture

This note proves Thomas Hermann's conjecture on the comparison between twoboundaries of the derivatives of rational cubic Bezier curves. The result is valuable for computer aided geometric design.     

A proof of Coleman's conjecture

Almost thirty years ago Coleman made a conjecture that for any convex lattice polygon with v vertices, g (g?1) interior lattice points and b boundary lattice points we have b?2g-v+10. In this note we give a proof of the conjecture. We also aim to describe all convex lattice polygons for which the bound b=2g-v+10 is attained.     

Short proof of a conjecture concerning split-by-nilpotent extensions

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of \(\mathop {\text {mod}}B\) inherited by \(\mathop {\text {mod}}C\). We show if B is a tilted algebra, then C is a tilted algebra.     

On the Nochka-Chen-Ru-Wong proof of Cartan's conjecture

In 1982-1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in Pn relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subsequently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides further simplification of the proof of the construction of the weights, by bringing back the use of the convex hull in working with the “Nochka diagram.”