Analytic parametrization and volume minimization of three dimensional bodies of constant width
نویسندگان
چکیده
We present a complete analytic parametrization of constant width bodies in dimension 3 based on the median surface: more precisely, we define a bijection between some space of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke’s formula. Finally, we derive weak optimality conditions for convex bodies which minimize the volume among constant width bodies.
منابع مشابه
Semidefinite programming for optimizing convex bodies under width constraints
We consider the problem of minimizing a functional (like the area, perimeter, surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of cons...
متن کاملMandibular Dimensional Changes with aging in Three Dimensional Computed Tomographic Study in 21 to 50 Year old Men and Women
Introduction: Raising the knowledge of skeletal and soft tissue changes with aging has been highly essential due to an increasing demand for aesthetic facial surgery following aging. The aim of this study is to evaluate the three dimensional computed tomographic images and process of changes in mandible with aging. Materials and Methods: In this descriptive study, the facial CT scans were obta...
متن کاملThe Blaschke-Lebesgue problem for constant width bodies of revolution
We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry. 2000 MSC: 52A15 Introduction The width of a convex body B in n-dimensional Euclidean space in the direction ~u is the distance between the two supporting planes of B ...
متن کاملNakajima’s Problem: Convex Bodies of Constant Width and Constant Brightness
For a convex body K ⊂ Rn, the kth projection function of K assigns to any k-dimensional linear subspace of Rn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in Rn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and ...
متن کاملIlluminating 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 con...
متن کامل