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  • Author or Editor: Endre Makai x
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Abstract  

The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ∈ Ob Ab | G is a torsion group, and for all gG the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, …,∞}. (o(·) means the order of an element, and n ≤ ∞ means n < ∞.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ∈ Ob CompAb | G has a neighbourhood subbase {G α} at 0, consisting of open subgroups, such that G/G α is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.

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K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let K be a convex body in the Euclidean plane such that any two boundary points p and q of K , that divide the circumference of K into two arcs of equal length, are antipodal. Then K is centrally symmetric. [19] announced the analogous result for any Minkowski plane

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, with arc length measured in the respective Minkowski metric. This was recently proved by Y. D. Chai — Y. I. Kim [7] and G. Averkov [4]. On the other hand, for Euclidean d -space ℝ d , R. Schneider [38] proved that if K ⊂ ℝ d is a convex body, such that each shadow boundary of K with respect to parallel illumination halves the Euclidean surface area of K (for the definition of “halving” see in the paper), then K is centrally symmetric. (This implies the result from [19] for ℝ 2 .) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let
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be a d -dimensional Minkowski space, and K
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be a convex body. If some Minkowskian surface area (e.g., Busemann’s or Holmes-Thompson’s) of K is halved by each shadow boundary of K with respect to parallel illumination, then K is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ: S d −1 → ℝ, for a convex body K with Euclidean surface area measure dS K ( u ), with ϕ( u ) being dS K ( u )-almost everywhere non-0, by the formula B ↦ ∫ B ϕ( u ) dS K ( u ) (supposing that ϕ is integrable with respect to dS K ( u )), for BS d −1 a Borel set, rather than the Euclidean surface area measure B ↦ ∫ B dS K ( u ). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure dS K ( u ) by the k -th area measure of K ( k with 1 ≦ kd − 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ d , which is due to Schneider [38].

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In a recent paper the authors have proved that a convex body K R d, d = 2, containing the origin O in its interior, is symmetric with respect to O if and only if V d - 1 (K \ H0) = V d - 1 (K \ H) for all hyperplanes H;H0 such that H and H0 are parallel and H0 3 O (V d - 1 is (d - 1){measure). For the proof the authors have employed a new type of integro-differential transform that lets to correspond to a suficiently nice function f on S d - 1 the function R (1) f, where(R (1) f)(.)= R S d - 1 \ . ? (f=)d {with. 2 S d - 1 as pole and as geographic latitude {and have determined the null-space of the operator R (1) . In this paper we extend the definition to any integer m = 1, defining (R (m) f)(.) analogously as for m=1, but using  m f= m rather than f= . (Thecasem=0 is the spherical Radon transformation (Funk transformation).) We investigate the null-space of the operator R (m) : up to a summand of finite dimension, it consists of the even (odd) functions in the domain of the operator, for m odd (even). For the proof we use spherical harmonics.

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High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C + 2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds).

Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds).

We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points S d for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

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