On the shadow boundary of a centrally symmetric convex body
نویسنده
چکیده
If K is a 0-symmetric, bounded, convex body in the Euclidean n-space R (with a fixed origin O) then it defines a norm whose unit ball is K itself (see [12]). Such a space is called Minkowski normed space. The main results in this topic collected in the survey [16] and [17]. In fact, the norm is a continuous function which is considered (in the geometric terminology as in [12]) as a gauge function. The metric (the so-called Minkowski metric), the distance of two points, induced by this norm, is invariant with respect to the translations of the space.
منابع مشابه
On Shadow Boundaries of Centrally Symmetric Convex Bodies
We discuss the concept of the so-called shadow boundary belonging to a given direction x of Euclidean n-space R lying in the boundary of a centrally symmetric convex body K. Actually, K can be considered as the unit ball of a finite dimensional normed linear (= Minkowski) space. We introduce the notion of the general parameter spheres of K corresponding to the above direction x and prove that i...
متن کاملCharacterizations of Central Symmetry for Convex Bodies in Minkowski Spaces
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 M2, with arc length measured in the respective Minkowsk...
متن کاملThe Unit Distance Problem for Centrally Symmetric Convex Polygons
Let f(n) be the maximum number of unit distances determined by the vertices of a convex n-gon. Erdős and Moser conjectured that this function is linear. Supporting this conjecture we prove that f (n) ∼ 2n where f (n) is the restriction of f (n) to centrally symmetric convex n-gons. We also present two applications of this result. Given a strictly convex domain K with smooth boundary, if fK (n) ...
متن کاملOn Boundary Arcs Joining Antipodal Points of a Planar Convex Body
Using notions of Minkowski geometry (i.e., of the geometry of finite dimensional Banach spaces) we find new characterizations of centrally symmetric convex bodies, equiframed curves, bodies of constant width and certain convex bodies with modified constant width property. In particular, we show that straightforward extensions of some properties of bodies of constant Euclidean width are also val...
متن کاملUPPER BOUNDS FOR THE COVERING NUMBER OF CENTRALLY SYMMETRIC CONVEX BODIES IN Rn
The covering number c(K) of a convex body K is the least number of smaller homothetic copies of K needed to cover K . We provide new upper bounds for c(K) when K is centrally symmetric by introducing and studying the generalized α -blocking number βα 2 (K) of K . It is shown that when a centrally symmetric convex body K is sufficiently close to a centrally symmetric convex body K′ , then c(K) i...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007