Results 251 to 260 of about 3,694,682 (291)
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Journal of the London Mathematical Society, 1994
The authors consider two operations, \(\square\) and \(\diamondsuit\), on the class of convex bodies. The first one is product (or direct sum) with Euclidean metric (compare the reviewer's paper in Glasnik Mat. 27(47), 145-158 (1992)). The second is defined by the formula: \(A\diamondsuit B=\text{conv}(A\times \{b\}\cup \{a\}\times B)\), where \(a ...
Farran, H. R., Robertson, S. A.
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The authors consider two operations, \(\square\) and \(\diamondsuit\), on the class of convex bodies. The first one is product (or direct sum) with Euclidean metric (compare the reviewer's paper in Glasnik Mat. 27(47), 145-158 (1992)). The second is defined by the formula: \(A\diamondsuit B=\text{conv}(A\times \{b\}\cup \{a\}\times B)\), where \(a ...
Farran, H. R., Robertson, S. A.
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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.
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Chasing Convex Bodies and Functions
Lecture Notes in Computer Science, 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.
Kirk Pruhs +2 more
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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.
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Journal of Convex Analysis, 2011
Everybody knows what an extreme point of a convex body is. Does this notion have an opposite? A most non-extreme point? Or: can we somehow say that a point is more or less extreme than another? In this paper we show a way to do this. We shall define the notion of moderation for points of a convex body.
Itoh, Jin-Ichi, Zamfirescu, Tudor
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Everybody knows what an extreme point of a convex body is. Does this notion have an opposite? A most non-extreme point? Or: can we somehow say that a point is more or less extreme than another? In this paper we show a way to do this. We shall define the notion of moderation for points of a convex body.
Itoh, Jin-Ichi, Zamfirescu, Tudor
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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.
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The cross-section body, plane sections of convex bodies and approximation of convex bodies, I
Geometriae Dedicata, 1996zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Makai, E. jun., Martini, H.
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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
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