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1.
Rolf Schneider 《Israel Journal of Mathematics》1971,9(2):241-249
Ifs is a mapping from the set of all convex bodies in Euclidean spaceE
d
toE
d
which is additive (in the sense of Minkowski), equivariant with respect to proper motions, and continuous, thens(K) is the Steiner point of the convex bodyK. 相似文献
2.
LetK be a convex body in a Euclideand-spaceE
d withd1. In 1957, H. Hadwiger conjectured thatK can always be covered by 2
d
smaller homothetic copies ofK. We verify this conjecture in the case thatK is the polar of a cyclicd-polytope andd=3, 4 and 5. 相似文献
3.
Daniel M. Oberlin 《Journal of Geometric Analysis》2010,20(2):422-438
We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝ
d
. 相似文献
4.
Every 2n-dimensional normed spaceE contains twon-dimensional subspacesE
1 andE
2 which are orthogonal with respect to the inner product induced by the John ellipsoid ofE and which satisfyd(E
i, l
2
n
)≦f(K
2(E)), wheref(K
2(E)) is some number that depends only on the cotype constant ofE, denotedK
2(E).
Supported in part by NSF grant DMS 8401906. 相似文献
5.
We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d
D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free.
The first author is supported by BFM2003-06092. 相似文献
6.
K. Bezdek 《Geometriae Dedicata》1993,45(1):89-91
LetKE
d
be a convex body and letl
r(K) denote the minimum number ofr-dimensional affine subspaces ofE
d
lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl
r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113. 相似文献
7.
LetM be a compact, convex set of diameter 2 inE
d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.
Mit 1 Abbildung 相似文献
Mit 1 Abbildung 相似文献
8.
G. T. Sallee 《Israel Journal of Mathematics》1976,24(3-4):368-376
LetK be a compact, convex subset ofE
dwhich can be tiled by a finite number of disjoint (on interiors) translates of some compact setY. Then we may writeK=X+Y, whereX is finite. The possible structures forK, X andY are completely determined under these conditions. 相似文献
9.
It is shown that a Banach space E has type p if and only for some (all) d ≥ 1 the Besov space B(1/p – 1/2)d p,p (?d ; E) embeds into the space γ (L2(?d ), E) of γ ‐radonifying operators L2(?d ) → E. A similar result characterizing cotype q is obtained. These results may be viewed as E ‐valued extensions of the classical Sobolev embedding theorems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
Bambah, Rogers, Woods, and Zassenhaus considered the general problem of covering planar convex bodiesC byk translates of a centrally-symmetric convex bodyK ofE
2 with the ramification that these translates cover the convex hullC
k
of their centres. They proved interesting inequalities for the volume ofC andC
k
. In the present paper some analogous results in euclideand-spaceE
d
are given. It turns out that on one hand extremal configurations ford5 are of quite different type than in the planar case. On the other hand inequalities similar to the planar ones seem to exist in general. Inequalities in both directions for the volume and other quermass-integrals are given. 相似文献
11.
Álvaro Lozano-Robledo 《The Ramanujan Journal》2009,19(1):53-61
In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E
d
,E
d′ and E
dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.
相似文献
12.
Given a setS ofn points inR
d
, a subsetX of sized is called ak-simplex if the hyperplane aff(X) has exactlyk points on one side. We studyE
d
(k,n), the expected number of k-simplices whenS is a random sample ofn points from a probability distributionP onR
d
. WhenP is spherically symmetric we prove thatE
d
(k, n)≤cn
d−1 WhenP is uniform on a convex bodyK⊂R
2 we prove thatE
2
(k, n) is asymptotically linear in the rangecn≤k≤n/2 and whenk is constant it is asymptotically the expected number of vertices on the convex hull ofS. Finally, we construct a distributionP onR
2 for whichE
2((n−2)/2,n) iscn logn.
The authors express gratitude to the NSF DIMACS Center at Rutgers and Princeton. The research of I. Bárány was supported in
part by Hungarian National Science Foundation Grants 1907 and 1909, and W. Steiger's research was supported in part by NSF
Grants CCR-8902522 and CCR-9111491. 相似文献
13.
Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non‐negative real‐valued function f : 𝒦→ ℝ+ is called a fractional K‐factor if ∑ K:v∈K∈𝒦f(K) ≤ 1 for every v ∈ V and ∑ K∈𝒦f(K) = n/r. For a non‐empty graph K let d(K) = e(K)/v(K) and d(1)(K) = e(K)/(v(K) ‐ 1). We say that K is strictly K1‐balanced if for every proper subgraph K′ ⊊ K, d(1)(K′) < d(1)(K). We say that K is imbalanced if it has a subgraph K′ such that d(K′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K1‐balanced, then with probability tending to 1 as n →∞, at the first moment τ0 when every vertex is covered by a copy of K, the graph has a fractional K‐factor. This result is the best possible. As a consequence, if K is K1‐balanced, we derive the threshold probability function for a random graph to have a fractional K‐factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ0 and the appearance of a fractional K‐factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 相似文献
14.
We disprove the longstanding conjecture that every combinatorial automorphism of the boundary complex of a convex polytope
in euclidean spaceE
d
can be realised by an affine transformation ofE
d
. 相似文献
15.
Imre Brny 《Random Structures and Algorithms》2007,30(3):414-426
Given a convex body K ⊂ R d, what is the probability that a randomly chosen congruent copy, K*, of K is lattice‐point free, that is, K*∩ Z d = ∅︁? Here Z d is the usual lattice of integer points in R d. Luckily, the underlying probability is well defined since integer translations of K can be factored out. The question came up in connection with integer programming. We explain what the answer is for convex bodies of large enough volume. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 相似文献
16.
J. -F. Le Gall 《Probability Theory and Related Fields》1988,78(3):389-402
Summary We consider the following heat conduction problem. Let K be a compact set in Euclidean space 3. Suppose that K is held at the temperature 1, while the surrounding medium is at the temperature 0 at time 0. Following Spitzer we investigate the asymptotic behaviour of the integral E
K
(t) which represents the total energy flow in time t from the set K to the surrounding medium 3–K. An asymptotic expansion is given for E
K
(t) which refines a theorem due to Spitzer. This expansion also verifies and improves a formal calculation of Kac. Similar results are proved in higher dimensions. Up to the constant m(K), the quantity E
K
(t) can be interpreted as the expected value of the volume of the Wiener sausage associated with K and a d-dimensional Brownian motion. This point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion. 相似文献
17.
Gerhard Frey 《Israel Journal of Mathematics》1994,85(1-3):79-83
Thed-th symmetric productC
(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies ifC
(d)
(K) is an infinite set, then there is aK-rational covering ofC → ℙ
|K
1
of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with [L∶K]≤d and withL-rational isogeny of degree ι is infinite. 相似文献
18.
We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE
d
(somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE
d
with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles. 相似文献
19.
Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures
Bang-Yen Chen 《Annals of Global Analysis and Geometry》2010,38(2):145-160
Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K
D
of a surface in Euclidean 4-space
\mathbb E4{\mathbb {E}^4} satisfy K + |K
D
| ≤ H
2, where H
2 is the squared mean curvature. A surface in
\mathbb E4{\mathbb {E}^4} is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in
\mathbb E4{\mathbb {E}^4} form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely
classify Wintgen ideal surfaces in
\mathbb E4{\mathbb E^4} satisfying |K| = |K
D
| identically. 相似文献
20.
Alexandr V. Kuzminykh 《Journal of Geometry》2004,79(1-2):134-145
A family of convex bodies in Ed is called neighborly if the
intersection of every two of them is (d-1)-dimensional. In the present paper we
prove that there is an infinite neighborly family of centrally symmetric convex bodies
in Ed, d 3, such that every two of them are affinely equivalent
(i.e., there is an affine transformation mapping one of them onto another), the
bodies have large groups of affine automorphisms, and the volumes of the bodies are
prescribed. We also prove that there is an infinite neighborly family of centrally
symmetric convex bodies in Ed such that the bodies have large groups of
symmetries. These two results are answers to a problem of B. Grünbaum (1963). We
prove also that there exist arbitrarily large neighborly families of similar convex
d-polytopes in Ed with prescribed diameters and with arbitrarily large
groups of symmetries of the polytopes. 相似文献