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L. E. Bazilevich 《Mathematical Notes》1997,62(6):683-687
The hyperspace of all convex bodies of constant width in Euclidean spaceR
n
,n≥2, is proved to be homeomorphic to a contractibleQ-manifold (Q denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex
bodies on the hyperspace of convex bodies of constant width.
Translated fromMaternaticheskie Zametki, Vol. 62, No. 6, pp. 813–819, December, 1997
Translated by V. N. Dubrovsky 相似文献
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Paulo Ventura Ara 《Geometriae Dedicata》1997,64(1):41-53
We prove that, in the hyperbolic plane, the Reuleaux triangle has smaller area than any other set of the same constant width. 相似文献
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Emanuel Milman 《Journal of Theoretical Probability》2009,22(1):256-278
Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s
quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power
type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L
p
for 1<p<∞. The same is true when L
p
is replaced by S
p
m
, the l
p
-Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration
of volume observation for uniformly convex bodies.
Supported in part by BSF and ISF. 相似文献
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该文首先定义了一类平面曲线"杠杆轮"与它的臂函数,并利用臂函数给出杠杆轮的参数表示.其次,证明了杠杆轮是平面常宽曲线的一种等价刻画.最后,表明Reuleaux多边形是臂函数为分段常函数的一类杠杆轮,进而构造出偶数边的Reuleaux多边形. 相似文献
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Elisabetta Maluta David Yost 《Journal of Mathematical Analysis and Applications》2019,469(2):1080-1087
We prove that every Banach space which admits an unconditional basis can be renormed to contain a constant width set with empty interior, thus guaranteeing, for the first time, existence of such sets in a reflexive space. In the isometric case we prove that normal structure is characterized by the property that the class of diametrically complete sets and the class of sets with constant radius from the boundary coincide. 相似文献
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We discuss the value of the best constant in Gaffney inequality namely when either or on ?Ω. 相似文献
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推广了Etzion和Vardy关于常维码的结论(Etzion T,Vardy A.Error-correcting codes in projective space.IEEE Transactions on Information Theory,2011,57(2):1165-1173),给出了一般情况下常维码的一个构造性下界. 相似文献
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凸体的曲率映象与仿射表面积 总被引:4,自引:0,他引:4
本文研究了一些特殊凸体与其极体的曲率仿射表面积乘积的下界.对任意两个凸体,建立了它们的投影体的混合体积与其仿射表面积的一个不等式(见文[1-15]). 相似文献
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Let K be an isotropic convex body in and let Zq(K) be the Lq-centroid body of K. For every N>n consider the random polytope KN:=conv{x1,…,xN} where x1,…,xN are independent random points, uniformly distributed in K. We prove that a random KN is “asymptotically equivalent” to Z[ln(N/n)](K) in the following sense: there exist absolute constants ρ1,ρ2>0 such that, for all and all NN(n,β), one has:
- (i) KNc(β)Zq(K) for every qρ1ln(N/n), with probability greater than 1−c1exp(−c2N1−βnβ).
- (ii) For every qρ2ln(N/n), the expected mean width of KN is bounded by c3w(Zq(K)).
Keywords: Convex body; Isotropic body; Isotropic constant; Random polytope; Centroid bodies; Mean width; Volume radius 相似文献
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In this paper, some new convergent sequences and inequalities of Euler's constant are provided. To demonstrate the superiority of our new convergent sequence over DeTemple's sequence, Vernescu's sequence and Mortici's sequences, some numerical computations are also given. 相似文献
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Qi Guo 《Discrete and Computational Geometry》2005,34(2):351-362
Given a convex body $C\subset R^n$ (i.e., a compact convex set with nonempty
interior), for $x\in$ {\it int}$(C)$, the interior, and a hyperplane $H$ with $x\in H$,
let $H_1,H_2$ be the two support hyperplanes of $C$ parallel to $H$. Let $r(H, x)$
be the ratio, not less than 1, in which $H$ divides the distance between
$H_1,H_2$. Then the quantity
$${\it As}(C):=\inf_{x\in {\it int}(C)}\,\sup_{H\ni x}\,r(H,x)$$
is called the Minkowski measure of asymmetry of $C$. {\it As}$(\cdot)$ can be viewed as a real-valued function defined on the family of
all convex bodies in $R^n$. It has been known for a long time that {\it As}$(\cdot)$
attains its minimum value 1 at all centrally symmetric convex bodies and maximum
value $n$ at all simplexes. In this paper we discuss the stability of the
Minkowski measure of asymmetry for convex bodies. We give an estimate for the
deviation of a convex body from a simplex if the corresponding Minkowski measure
of asymmetry is close to its maximum value. More precisely, the following result
is obtained: Let $C\subset R^n$ be a convex body. If {\it As}$(C)\ge n-\varepsilon$ for some $0\le
\varepsilon < 1/8(n+1),$ then there exists a simplex $S_0$ formed by $n+1$
support hyperplanes of $C$, such that
$$(1+8(n+1)\varepsilon)^{-1}S_0\subset C\subset S_0,$$
where the homethety center is the (unique) Minkowski critical point of $C$. So
$$d_{{\rm BM}}(C,S)\le 1+8(n+1)\varepsilon$$
holds for all simplexes $S$, where $d_{{\rm BM}}(\cdot,\cdot)$ denotes the Banach-Mazur distance. 相似文献
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通过引入刻画平面常宽凸域的不对称性函数,证明了在平面常宽凸域中,圆域 是最对称的,而Reuleaux三角形是最不对称的. 相似文献
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Azmy S. Ackleh Sergiu Aizicovici Simeon Reich 《Numerical Functional Analysis & Optimization》2013,34(5-6):553-570
We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed. 相似文献
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Chunije Wang 《Mathematische Nachrichten》2008,281(3):447-454
Let Ap (??) (p ≥ 1) be the Bergman space over the open unit disk ?? in the complex plane. Korenblum's maximum principle states that there is an absolute constant c ∈ (0, 1) (may depend on p), such that whenever |f (z)| ≤ |g (z)| (f, g ∈ Ap (??)) in the annulus c < |z | < 1, then ∥f ≤ ∥g ∥. For p ≥ 1, let cp be the largest value of c for which Korenblum's maximum principle holds. In this note we prove that cp → 1 as p → ∞. Thus we give a positive answer of a question of Hinkkanen. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Saba NAZ Muhammad SALMAN Usman ALI Imran JAVAID Syed Ahtsham-ul-Haq BOKHARY 《数学学报(英文版)》2014,30(7):1145-1160
In this paper,we consider the family of generalized Petersen graphs P(n,4).We prove that the metric dimension of P(n,4) is 3 when n ≡ 0(mod 4),and is 4 when n = 4k + 3(k is even).For n ≡ 1,2(mod 4) and n = 4k + 3(k is odd),we prove that the metric dimension of P(n,4) is bounded above by 4.This shows that each graph of the family of generalized Petersen graphs P(n,4)has constant metric dimension. 相似文献
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Geoffrey R. Grimmett Thomas M. Liggett Thomas Richthammer 《Random Structures and Algorithms》2010,37(1):85-99
We consider a type of long‐range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and we provide partial results in others. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010 相似文献