On polars of mixed projection bodies

Recently, Lutwak established general Minkowski inequality, Brunn-Minkowski inequality and Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we established their polar forms. As applications, we prove some interrelated results.     

Convex polytopes all of whose reverse lexicographic initial ideals are squarefree

A compressed polytope is an integral convex polytope any of whose reverse lexicographic initial ideals is squarefree. A sufficient condition for a -polytope to be compressed will be presented. One of its immediate consequences is that the class of compressed -polytopes includes (i) hypersimplices, (ii) order polytopes of finite partially ordered sets, and (iii) stable polytopes of perfect graphs.

     

On zonoids whose polars are zonoids

Zonoids whose polars are zonoids cannot have proper faces other than vertices or facets. However, there exist non-smooth zonoids whose polars are zonoids. Examples in ℝ and ℝ are given. Supported in part by the United States-Israel Binational Science Foundation.     

Inequalities for polars of mixed projection bodies

In 1993 Lutwak established some analogs of the Brunn-Minkowsi inequality and the Aleksandrov-Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we give their polars forms. Further, as applications of our methods, we give a generalization of Pythagorean inequality for mixed volumes.     

Convex surfaces whose geodesics are planar

It is shown that ifC is a convex surface, in euclidean space of dimension at least 3, having the property that all shortest paths onC between pairs of its points are planar, thenC is a sphere, a hyperplane or the boundary of an intersection of two half-spaces. No smoothness assumptions are made.     

Convex polytopes and formal groups

Some constructions of commutative formal groups proceeding from convex polytopes and Laurent polynomials are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 87–90, 1978.     

Semigroups whose Julia sets are Cantor targets

We consider semigroups generated by two rational functions whose Julia sets are Cantor targets. Noting that a Cantor target has no interior points, we construct a polynomial semigroup whose Julia set has no interior points and the Hausdorff dimension of whose Julia set is arbitrary close to 2.     

Non-intersection bodies, all of whose central sections are intersection bodies

We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of .

     

Convex sets which are intersections of closed balls

In this paper we aim to investigate different questions concerning the stability of the set of all intersections of closed balls in a normed space. We are mainly concerned with: (i) the stability of under the closure of the vector sums; (ii) the stability under the addition of balls. We prove that (i) and (ii) are different properties which have strong connections with the geometry of the space. They have interest both in finite and infinite dimension. In the former case, there is a link with linear programming theory. We also study two more stability properties related to the well-known binary intersection property. Mazur sets and Mazur spaces are introduced, as a natural family satisfying (i). We prove that every two-dimensional normed space is a Mazur space, a result which distinguishes dimension d?2 from dimension d?3. We also discuss the connections between Mazur spaces and porosity.     

Convex polyhedra whose faces are equiangular or composed of such

Depending on the type, considering only the topological structure of the network of faces, and the angles of corresponding faces at corresponding vertices, convex polyhedra in R³, each face of which is equiangular or composed of such, constitute four infinite series (prism, antiprism, and two types of truncated antiprisms); outside of this series, there are only a finite number of types. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 111–112, 1974.     

Non-measurable sets of reals whose measurable subsets are countable

In this paper we generalize Sierpinski's concept of sets of typeS and give a characterization of such sets in terms of a partition of the reals. We also give a similar characterization of Lusin sets.     

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,…,N}, chosen randomly according to a binomial model with parameter p(N), with N^?1 = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N^?1/2) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N^?1/2 = o(p(N)) then both the sum and difference sets almost surely have size (2N + 1) ? O(p(N)^?2), and so the ratio in question is almost surely very close to one. If p(N) = c · N^?1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_f,g · N^?1/2, for some computable constant c_f,g, such that one form almost surely dominates below the threshold and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Representations of polytopes and polyhedral sets

In this paper we describe a variant of the diagram techniques, such as Gale diagrams for polytopes and positive diagrams for positive bases, which is appropriate for polyhedral sets. We obtain our new technique as a translation-invariant representation of polytopes or polyhedral sets. This approach leads naturally to simpler proofs of the familiar combinatorial diagram relationships. However, this method is more versatile than those previously employed, in that it can be used to investigate linear systems of polyhedral sets, and metrical properties such as volume. In particular, we give an easy proof of a result of Meyer on decomposability of polytopes, and a more perspicuous way of looking at the well-known theorem of Minkowski on the realizability of polytopes whose facets have given outer normal vectors and areas.     

On graphs whose star sets are (co-)cliques

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G=K_1,2 or K_2,…,2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets.     

A class of functions whose sublevel sets are absolute retracts

In this paper, we prove a result of which the following is a corollary: If X is a Banach space and J:X→R is a contraction, then the nonempty sublevel sets of the function x→‖x‖+J(x) are absolute retracts.