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1.
Rolf Schneider 《Monatshefte für Mathematik》2007,150(3):241-247
If two convex bodies have the property that their orthogonal projections on any hyperplane have the same mean width and the
same Steiner point, then the bodies are identical. This result is proved in a stronger stability version. 相似文献
2.
The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of . We determine this constant and the similarly defined constant for approximation with a given number of facets by solving
two isoperimetric problems for planar tilings.
Received 15 May 1997; in revised form 14 August 1997 相似文献
3.
For 1 ≤ i < j < d, a j-dimensional subspace L of
and a convex body K in
, we consider the projection K|L of K onto L. The directed projection function v
i,j
(K;L,u) is defined to be the i-dimensional size of the part of K|L which is illuminated in direction u ∈ L. This involves the i-th surface area measure of K|L and is motivated by Groemer’s [17] notion of semi-girth of bodies in
. It is well-known that centrally symmetric bodies are determined (up to translation) by their projection functions, we extend
this by showing that an arbitrary body is determined by any one of its directed projection functions. We also obtain a corresponding
stability result. Groemer [17] addressed the case i = 1, j = 2, d = 3. For j > 1, we then consider the average of v
1,j
(K;L,u) over all spaces L containing u and investigate whether the resulting function
determines K. We will find pairs (d,j) for which this is the case and some pairs for which it is false. The latter situation will be seen to be related to some
classical results from number theory. We will also consider more general averages for the case of centrally symmetric bodies.
The research of the first author was supported in part by NSF Grant DMS-9971202 and that of the second author by a grant from
the Volkswagen Foundation. 相似文献
4.
For a given convex body K in
with C
2 boundary, let P
c
n
be the circumscribed polytope of minimal volume with at most n edges, and let P
i
n
be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P
c
n
and P
i
n
are asymptotically regular triangles and squares, respectively, in a suitable sense.
Supported by OTKA grants 043520 and 049301, and by the EU Marie Curie grants Discconvgeo, Budalggeo and PHD.
Authors’ addresses: Károly J. B?r?czky, Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest H–1364, Hungary, and
Department of Geometry, Roland E?tv?s University, Pázmány Péter sétány 1/C, Budapest 1117, Hungary; Salvador S. Gomis, Department
of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain; Péter Tick, Gyűrű utca 24, Budapest H–1039, Hungary 相似文献
5.
Mark W. Meckes 《Monatshefte für Mathematik》2005,145(4):307-319
We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms. 相似文献
6.
Manfred Peter 《Monatshefte für Mathematik》2002,135(1):37-57
An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to
the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the
second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds.
Received 25 April 2001 相似文献
7.
August Florian 《Monatshefte für Mathematik》2001,133(2):111-129
In this paper we provide an upper bound to the density of a packing of circles on the sphere, with radii selected from a
given finite set. This bound is precise, e.g. for the system of incircles of Archimedean tilings (4, 4, n) with n ? 6. A generalisation to the weighted density of packing is applied to problems of solidity of a packing of circles. The
simple concept of solidity was introduced by L. Fejes Toóth [6]. In particular, it is proved that the incircles of the faces of the Archimedean tilings
(4,6,6), (4,6,8) and (4, 6, 10) form solid packings.
(Received 21 August 2000; in revised form 21 March 2001) 相似文献
8.
Ekkehard Krätzel 《Monatshefte für Mathematik》2002,137(3):197-211
In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated
points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential
planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian
curvature zero on the estimation of the lattice rest.
Received 19 June 2001; in revised form 17 January 2002
RID="a"
ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday 相似文献
9.
We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.This paper was written while the third author was supported by a grant from DFG and TÜBITAK.Received March 12, 2002
Published online November 18, 2002 相似文献
10.
Wolfgang Müller 《Monatshefte für Mathematik》1999,128(4):315-330
Denote by the number of points of the lattice in the “blown up” domain , where is a convex body in () whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed
where for and . This improves a classic result of E. Hlawka [8] and its refinements due to E. Kr?tzel and W. G. Nowak ([14], [15]). The proof uses a multidimensional variant of the method of van der Corput for the estimation of exponential
sums.
Received 28 August 1998 相似文献
11.
Károly Böröczky Jr. María A. Hernández Cifre G. Salinas 《Monatshefte für Mathematik》2003,138(2):95-110
In this paper two problems posed by Santaló are solved: we determine the planar convex sets which have maximum and minimum
area or perimeter when the circumradius and the inradius are given, obtaining complete systems of inequalities for the cases
(A, R, r) and (p, R, r).
This work is supported in part by Dirección General de Investigación (MCYT) BFM2001-2871, and by OTKA grants No 31984 and
30012
Received October 15, 2001; revised January 29, 2002 相似文献
12.
We prove that if a convex body
has an interior false pole with respect to some hyperplane, then the body is an ellipsoid.
This research was partially carried out during the postdoctoral visit of this author at University College London, and it
was supported by CONACYT, México. 相似文献
13.
Ekkehard Krätzel 《Monatshefte für Mathematik》2004,143(2):145-162
We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero. 相似文献
14.
An extremal property of the mean width of the simplex 总被引:4,自引:0,他引:4
Franck Barthe 《Mathematische Annalen》1998,310(4):685-693
15.
We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces.
Authors’ addresses: Mirjana Djorić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade,
Serbia; Masafumi Okumura, 5-25-25 Minami Ikuta, Tama-ku, Kawasaki, Japan 相似文献
16.
Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.Research supported by a grant from a cooperation between the Deutsche Forschungsgemeinschaft in Germany and the National Research Foundation in South Africa. Parts of this paper were written during a visit to the Department of Mathematics, Applied Mathematics and Astronomy of the University of South Africa. 相似文献
17.
In this paper, we are interested in extending the study of spherical curves in R
3 to the submanifolds in the Euclidean space R
n+p
. More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R
n+p
lies on the hypersphere S
n+p−1(c) (standardly imbedded sphere in R
n+p
of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold
(cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S
n
(c) (cf. Theorem 4.1). 相似文献
18.
Martin Wiehe 《Mathematische Zeitschrift》2002,241(2):353-373
We develop a unimodularly invariant theory for immersions with higher codimension into the affine space.
Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002
RID="*"
ID="*" Supported by the Deutsche Forschungsgemeinschaft 相似文献
19.
In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space
containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the
only extremal bodies, which is proved for all three dimensional and several prominent lattices.
Authors’ addresses: M. A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia,
Spain; A. Schürmann, Institut für Algebra und Geometrie, Otto-von-Guericke Universit?t Magdeburg, 39106 Magdeburg, Germany;
F. Vallentin, Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands 相似文献
20.
August Florian 《Monatshefte für Mathematik》2007,152(1):39-43
The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a
given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note
it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof
of the theorem. 相似文献