平面常宽凸域的不对称性度量

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．     

Asymmetry of Convex Bodies of Constant Width

The symmetry of convex bodies of constant width is discussed in this paper. We proved that for any convex body K?? ⁿ of constant width, \(1\leq \mathrm{as}_{\infty}(K)\leq\frac{n+\sqrt{2n(n+1)}}{n+2}\), where as_∞(?) denotes the Minkowski measure of asymmetry for convex bodies. Moreover, the equality holds on the left-hand side precisely iff K is an Euclidean ball and the upper bounds are attainable, in particular, if n=3, the equality holds on the right-hand side if K is a Meissner body.     

Measures of Asymmetry Dual to Mean Minkowski Measures of Asymmetry for Convex Bo dies

We introduce a family of measures (functions) of asymmetry for convex bodies and discuss their properties. It turns out that this family of measures shares many nice properties with the mean Minkowski measures. As the mean Minkowski measures describe the symmetry of lower dimensional sections of a convex body, these new measures describe the symmetry of lower dimensional orthogonal projections.     

On the Asymmetry Constant of a Body with Few Vertices

In this note, we show that a nondegenerated polytope in IR ⁿ with n + k, 1 k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes.     

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

On Prop erties of p-critical Points of Convex Bo dies

Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on(1, +∞) is confirmed, and the connections between general p-critical points and the Minkowski-critical points(∞-critical points) are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.     

Stability of the Minkowski Measure of Asymmetry for Convex Bodies

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.     

Approximation of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.     

Minimum Area of a Set of Constant Width in the Hyperbolic Plane

We prove that, in the hyperbolic plane, the Reuleaux triangle has smaller area than any other set of the same constant width.     

Surfaces of Constant Mean Curvature Bounded by Convex Curves

This paper proves that an embedded compact surface in the Euclidean space with constant mean curvature H bounded by a circle of radius 1 and included in a slab of width is a spherical cap. Also, we give partial answers to the problem when a surface with constant mean curvature and planar boundary lies in one of the halfspaces determined by the plane containing the boundary, exactly, when the surface is included in a slab.     

Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating convex bodies of constant width to the family of “fat” spindle convex bodies. Also, this leads to the spherical analog of the well-known Blaschke–Lebesgue problem.     

Hilbert Geometry for Strictly Convex Domains

We prove in this paper that the Hilbert geometry associated with a bounded open convex domain in R ⁿ whose boundary is a ² hypersuface with nonvanishing Gaussian curvature is bi-Lipschitz equivalent to the n-dimensional hyperbolic space H ⁿ . Moreover, we show that the balls in such a Hilbert geometry have the same volume growth entropy as those in H ⁿ .     

关于具有常平均曲率的超曲面

以N表示其截面曲率KN满足a≤KN≤b的n＋1维单连通完备黎曼流形，Mn是Nn 1中具有常平均曲率的紧致超曲面，本文给出了这类超曲面关于其第二基本形式模长平方S的积分不等式及相应的刚性定理．     

Curvature Estimates for Strongly Stable Domains on Surfaces with Constant Mean Curvature in Space Forms

ＣｕｒｖａｔｕｒｅＥｓｔｉｍａｔｅｓｆｏｒＳｔｒｏｎｇｌｙＳｔａｂｌｅＤｏｍａｉｎｓｏｎＳｕｒｆａｃｅｓｗｉｔｈＣｏｎｓｔａｎｔＭｅａｎＣｕｒｖａｔｕｒｅｉｎＳｐａｃｅＦｏｒｍｓ（ＷａｎｇＺｈｉｇｕｏ）；（王治国）（ＷｕＺｈｉｑｉｎｇ）；（吴志勤）（...     

Probabilistic Adaptive Width of a Multivariate Sobolev Space Equipped with a Gaussian Measure

In this paper, we determine the asymptotic values of the probabilistic adaptive widths of the space of multivariate functions with bounded mixed derivative (MW₂^r(T^d),μ) relative to the manifold (Y^N,ν) in the L_q(T^d)-norm, 1 < q ≤ 2, where μ and ν are two given Gaussian measures.     

Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: -convexity (in the sense of Klötzler and Hartwig), -convexity and midpoint -convexity (in the sense of Hu, Klee, and Larman), -convexity and midpoint -convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.