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1. CJM Online first

Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros
Inequalities for the surface area of projections of convex bodies
We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Keywords:surface area, convex body, projection
Categories:52A20, 46B05

2. CJM Online first

Cordero-Erausquin, Dario
Transport inequalities for log-concave measures, quantitative forms and applications
We review some simple techniques based on monotone mass transport that allow us to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.

Keywords:log-concave measures, transport inequality, Brascamp-Lieb inequality, quantitative inequalities
Categories:52A40, 60E15, 49Q20

3. CJM Online first

Choi, Suyoung; Park, Hanchul
Wedge operations and torus symmetries II
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. We refer to $K$ as a seed if $K$ cannot be obtained by wedgings. First, we show that, for a fixed positive integer $\ell$, there are at most finitely many seeds of Picard number $\ell$ supporting characteristic maps. As a corollary, the conjecture proposed by V.V. Batyrev in 1991 is solved affirmatively. Second, we investigate a systematic method to find all characteristic maps on $K(J)$ using combinatorial objects called (realizable) puzzles that only depend on a seed $K$. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Keywords:puzzle, toric variety, simplicial wedge, characteristic map
Categories:57S25, 14M25, 52B11, 13F55, 18A10

4. CJM 2016 (vol 68 pp. 762)

Colesanti, Andrea; Gómez, Eugenia Saorín; Nicolás, Jesus Yepes
On a Linear Refinement of the Prékopa-Leindler Inequality
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the $0$-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

Keywords:Prékopa-Leindler inequality, linearity, Asplund sum, projections, Borell-Brascamp-Lieb inequality
Categories:52A40, 26D15, 26B25

5. CJM Online first

Xiao, Jie; Ye, Deping
Anisotropic Sobolev Capacity with Fractional Order
In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha\rightarrow 0^+$, will be provided.

Keywords:sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeter
Categories:52A38, 53A15, 53A30

6. CJM Online first

De Bernardi, Carlo Alberto; Veselý, Libor
Tilings of normed spaces
By a tiling of a topological linear space $X$ we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite-dimensional spaces initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following. 1. $X$ admits no tiling by Fréchet smooth bounded tiles. 2. If $X$ is locally uniformly rotund (LUR), it does not admit any tiling by balls. 3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$ uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Keywords:tiling of normed space, Fréchet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$-space
Categories:46B20, 52A99, 46A45

7. CJM 2016 (vol 68 pp. 876)

Ostrovskii, Mikhail; Randrianantoanina, Beata
Metric Spaces Admitting Low-distortion Embeddings into All $n$-dimensional Banach Spaces
For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that $n$-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G. Schechtman.

Keywords:basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametric
Categories:46B85, 05C12, 30L05, 46B15, 52A21

8. CJM Online first

Cordero-Erausquin, Dario
Transport inequalities for log-concave measures, quantitative forms and applications
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure. We discuss quantitative forms of these inequalities, with application to the variance Brascamp-Lieb inequality.

Keywords:log-concave measures, transport inequality, Brascamp-Lieb inequality, quantitative inequalities
Categories:52A40, 60E15, 49Q20

9. CJM 2013 (vol 67 pp. 3)

Alfonseca, M. Angeles; Kim, Jaegil
On the Local Convexity of Intersection Bodies of Revolution
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

Keywords:convex bodies, intersection bodies of star bodies, Busemann's theorem, local convexity
Categories:52A20, 52A38, 44A12

10. CJM 2013 (vol 66 pp. 783)

Izmestiev, Ivan
Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional
The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity. We review some of the related work and discuss directions for future research.

Keywords:convex polyhedron, rigidity, Hilbert-Einstein functional, Minkowski theorem
Categories:52B99, 53C24

11. CJM 2012 (vol 65 pp. 1236)

De Bernardi, Carlo Alberto
Higher Connectedness Properties of Support Points and Functionals of Convex Sets
We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.

Keywords:convex set, support point, support functional, absolute retract, Leray-Schauder continuation principle
Categories:46A55, 46B99, 52A07

12. CJM 2012 (vol 66 pp. 700)

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

13. CJM 2012 (vol 65 pp. 675)

Strungaru, Nicolae
On the Bragg Diffraction Spectra of a Meyer Set
Meyer sets have a relatively dense set of Bragg peaks and for this reason they may be considered as basic mathematical examples of (aperiodic) crystals. In this paper we investigate the pure point part of the diffraction of Meyer sets in more detail. The results are of two kinds. First we show that given a Meyer set and any positive intensity $a$ less than the maximum intensity of its Bragg peaks, the set of Bragg peaks whose intensity exceeds $a$ is itself a Meyer set (in the Fourier space). Second we show that if a Meyer set is modified by addition and removal of points in such a way that its density is not altered too much (the allowable amount being given explicitly as a proportion of the original density) then the newly obtained set still has a relatively dense set of Bragg peaks.

Keywords:diffraction, Meyer set, Bragg peaks

14. CJM 2012 (vol 65 pp. 1401)

Zhao, Wei; Shen, Yibing
A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results
In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger-Kazdan inequality and Santaló's formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

Keywords:Finsler manifold, Berger-Kazdan inequality, Berger-Kazdan comparison theorem, Santaló's formula, Croke's isoperimetric inequality
Categories:53B40, 53C65, 52A38

15. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords:extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
Categories:42B05, 11T24, 52C17

16. CJM 2011 (vol 63 pp. 1254)

D'Azevedo, Antonio Breda; Jones, Gareth A.; Schulte, Egon
Constructions of Chiral Polytopes of Small Rank
An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks $3$, $4$, and $5$.

Keywords:abstract regular polytope, chiral polytope, chiral maps
Categories:51M20, 52B15, 05C25

17. CJM 2011 (vol 63 pp. 1038)

Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
Critical Points and Resonance of Hyperplane Arrangements
If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements.

Keywords:hyperplane arrangement, master function, resonant weights, critical set
Categories:32S22, 55N25, 52C35

18. CJM 2011 (vol 63 pp. 1220)

Baake, Michael; Scharlau, Rudolf; Zeiner, Peter
Similar Sublattices of Planar Lattices
The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

Categories:11H06, 11R11, 52C05, 82D25

19. CJM 2010 (vol 62 pp. 1293)

Kasprzyk, Alexander M.
Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties.

Keywords:toric, Fano, threefold, canonical singularities, convex polytopes
Categories:14J30, 14J30, 14M25, 52B20

20. CJM 2010 (vol 62 pp. 1228)

Ardila, Federico; Fink, Alex; Rincón, Felipe
Valuations for Matroid Polytope Subdivisions
We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

Categories:05B35, 52B40, 52B45, 52C22

21. CJM 2010 (vol 62 pp. 975)

Bjorndahl, Christina; Karshon, Yael
Revisiting Tietze-Nakajima: Local and Global Convexity for Maps
A theorem of Tietze and Nakajima, from 1928, asserts that if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex, then it is convex. We give an analogous ``local to global convexity" theorem when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map from a topological space $X$ to $\mathbb{R}^n$ that satisfies certain local properties. Our motivation comes from the Condevaux--Dazord--Molino proof of the Atiyah--Guillemin--Sternberg convexity theorem in symplectic geometry.

Categories:53D20, 52B99

22. CJM 2010 (vol 62 pp. 1404)

Saroglou, Christos
Characterizations of Extremals for some Functionals on Convex Bodies
We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantity $$ \int_{x_0\in K}\cdots\int_{x_n\in K}[V(\textrm{conv}\{x_0,\dots,x_n\})]^pdx_0\cdots dx_n , n\geq d, p\geq 1 $$ is maximized by triangles among all planar convex bodies $K$ (parallelograms in the symmetric case). We show that these are the only maximizers, a fact proven by Giannopoulos for $p=1$. Moreover, if $h$: $\mathbb{R}_+\rightarrow \mathbb{R}_+$ is a strictly increasing function and $W_j$ is the $j$-th quermassintegral in $\mathbb{R}^d$, we prove that the functional $$ \int_{x_0\in K_0}\cdots\int_{x_n\in K_n}h(W_j(\textrm{conv}\{x_0,\dots,x_n\}))dx_0\cdots dx_n , n \geq d $$ is minimized among the $(n+1)$-tuples of convex bodies of fixed volumes if and only if $K_0,\dots,K_n$ are homothetic ellipsoids when $j=0$ (extending a result of Groemer) and Euclidean balls with the same center when $j>0$ (extending a result of Hartzoulaki and Paouris).

Categories:52A40, 52A22

23. CJM 2009 (vol 61 pp. 1300)

Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c
Monodromy Groups and Self-Invariance
For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure $\mathcal{P}^d$. When $\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$\nobreakdash-auto\-morphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Keywords:maps, abstract polytopes, self-duality, monodromy groups, medials of polyhedra
Categories:51M20, 05C25, 05C10, 05C30, 52B70

24. CJM 2009 (vol 61 pp. 904)

Saliola, Franco V.
The Face Semigroup Algebra of a Hyperplane Arrangement
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Categories:52C35, 05E25, 16S37

25. CJM 2009 (vol 61 pp. 888)

Novik, Isabella; Swartz, Ed
Face Ring Multiplicity via CM-Connectivity Sequences
The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjecture is also established for the face rings of all doubly Cohen--Macaulay complexes whose 1-skeleton's connectivity does not exceed the codimension plus one as well as for all $(d-1)$-dimensional $d$-Cohen--Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring $\field[\Delta]$ via the Cohen--Macaulay connectivity of the skeletons of $\Delta$.

Categories:13F55, 52B05;, 13H15;, 13D02;, 05B35
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