Given a graph G and a subset W ⊆ V (G), a Steiner W -tree is a tree of minimum order that contains all of W . Let S(W ) denote the set of all vertices in G that lie on some Steiner W -tree; we call S(W ) the Steiner interval of W . If S(W ) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a short...

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v, k,1) designs, mainly for t = 2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t > 2 most of these characterizations have rema...

A loop satisfies Moufang’s theorem whenever the subloop generated by three associating elements is a group. Moufang loops (loops that satisfy the Moufang identities) satisfy Moufang’s theorem, but it is possible for a loop that is not Moufang to nevertheless satisfy Moufang’s theorem. Steiner loops that are not Moufang loops are known to arise from Steiner triple systems in which some triangle ...

Let G = (V, E) be a plane graph with nonnegative edge weights, and letN be a family of k vertex sets N1, N2, . . . , Nk ⊆ V , called nets. Then a noncrossing Steiner forest forN in G is a set T of k trees T1, T2, . . . , Tk in G such that each tree Ti ∈ T connects all vertices, called terminals, in net Ni , any two trees in T do not cross each other, and the sum of edge weights of all trees is ...

The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets.

In this paper, we demonstrate the existence of iterated Steiner symmetrizations [see formula in PDF] that does not converge, even if sequence directions is dense unit sphere.

We use some basic results and ideas from the integral geometry to study certain properties of group codes. The properties being studied are generalized weights and spectra of linear block codes over a finite field and their analogues for lattice sphere packings in Euclidean space. We do not obtain any new results about linear codes, although several short and simple proofs for known results are...

In this paper, we prove the monotonicity of anisotropic perimeter sets finite under Steiner symmetrization by a variational formula volume and an inequality for lower outer Minkowski content. As consequence, give more direct proof Wulff symmetrization.

A set of points in a Steiner triple system (STS(v)) is said to be independent if no three of these points occur in the same block. In this paper we derive for each k ≤ 8 a closed formula for the number of independent sets of cardinality k in an STS(v). We use the formula to prove that every STS(21) has an independent set of cardinality eight and is as a consequence 4-colourable. AMS classificat...

These notes provide a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Streamlined derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and th...