Results 261 to 270 of about 3,482,591 (285)
Some of the next articles are maybe not open access.
Convex bodies with non‐convex cross‐section bodies
Mathematika, 1999For \(K\) a convex body in \(\mathbb{R}^d\), the inner \((d-1)\)-quermass at direction \(u\in S^{d-1}\) is defined by \[ m_K(u)= \max_{t\in R}\lambda_{d-1} \bigl( K\cap (u^\perp+ tu)\bigr), \] where \(\lambda_{d-1}\) denotes the \((d-1)\)-dimensional Lebesgue measure. The cross-section body \(CK\) of \(K\), introduced by H.
openaire +2 more sources
On the Minimal Convex Shell of a Convex Body
Canadian Mathematical Bulletin, 1993AbstractFor any convex body C in ℝd we introduce the notion of the convex shell and we prove that there exists a unique "minimal" convex shell, extending the notion of the minimal spherical shell of C. Then we prove that a "typical" convex body touches the boundary of its minimal convex shell in precisely d + 2 points.
openaire +2 more sources
Covering convex bodies by translates of convex bodies
Mathematika, 1997Hadwiger conjectured that the smallest number of translates of a convex body \(K\) required to cover \(K\) is \(2^n\). Here a number of known weaker estimates of the number of required translates, or lattice translates, are obtained as consequences of two simple results.
Rogers, C. A., Zong, C.
openaire +2 more sources
On Bodies Associated with a Given Convex Body
Canadian Mathematical Bulletin, 1996AbstractLet d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J.
Makai, Endre jun., Martini, Horst
openaire +2 more sources
Intersections of Convex Bodies
Journal of Mathematical Sciences, 2001For convex bodies \(K_0,K_1,\dots,K_m \subset\mathbb{R}^n\) and for \(\rho=(r_1, \dots,r_n) \in(\mathbb{R}^n)^m\), let \(\Phi(\rho): =K_0\cap \bigcup^m_{i =1} (K_1+r_i)\) and \(D:=\{\rho: \Phi(\rho) \neq\emptyset\}\). Then \(D\) is convex, and the family \(\{\Phi(\rho)\}_{\rho\in D}\) is concave.
openaire +1 more source
Convex bodies and convexity on grassmann cones (X): Projection functions of parallel convex bodies
Annali di Matematica Pura ed Applicata, 1965It is shown that if K is a convex polyhedron or a smooth convex body, then for sufficiently large positive ρ, the body parallel to K at distance ρ has convex projection functions. An example is given of a convex body which does not have this property.
Busemann, H., Shephard, Geoffrey C.
openaire +2 more sources
Chasing Convex Bodies and Functions
2016We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing. In Online Convex Optimization the input is an online sequence of convex functions over some Euclidean space. In response to a function, the online algorithm can move to any destination point in the Euclidean space.
Antoniadis, Antonios +5 more
openaire +2 more sources
Projected Area of Convex Bodies
Nature, 1948IN connexion with photometric determination of specific surface of finely divided material, it is essential to know the relation between the projected area of particles, which is actually measured, and the corresponding surface area. The solution of this problem was first given by A.
openaire +3 more sources
Convex Bodies Associated with a Given Convex Body
Journal of the London Mathematical Society, 1958Rogers, C. A., Shephard, Geoffrey C.
openaire +1 more source