A player moving in the plane is given a sequence of instructions of the following type: at step i a planar convex set Fi is specified, and the player has to move to a point in Fi. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequence Fi the cost to the player is within a constant (multiplicative) factor of the "off-lin...

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...

‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

Introduction. We propose to investigate here the consequences of the identity of each pair chosen from three important generalizations of the relation of betweenness on a line, namely, algebraic betweenness [l, p. 27 J, metric betweenness [3, p. 36], and lattice betweenness [7, Part I I ] . We shall also find an interpretation of metric betweenness in the Banach space of all continuous function...