We express the set of exposed points in terms of rotund points and non-smooth points.As long as we have Banach spaces each time＂bigger＂,we consider sets of non-smooth points each time＂smaller＂.
In this paper we derive sufficient conditions for strict convexity of subsets in a complete simply connected smooth Riemanian manifold without focal points in terms of local and global exposed points.
Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth p...Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth points which are also non-exposed.展开更多
In this paper,we prove that(X,p)is separable if and only if there exists a w^(*)-lower semicontinuous norm sequence{p_(n)}_(n=1)^(∞)of(X^(*),p)such that(1)there exists a dense subset G_(n)of X^(*)such that p_(n)is Ga...In this paper,we prove that(X,p)is separable if and only if there exists a w^(*)-lower semicontinuous norm sequence{p_(n)}_(n=1)^(∞)of(X^(*),p)such that(1)there exists a dense subset G_(n)of X^(*)such that p_(n)is Gateaux differentiable on G_(n)and dp_(n)(Gn_(n))■X for all n∈N;(2)p_(n)≤p and p_(n)→p uniformly on each bounded subset of X^(*);(3)for anyα∈(0,1),there exists a ball-covering{B(x^(*)i,n,Ti,n)}∞i=1 of(X^(*),p_(n))such that it isα-off the origin and x_(i,n)^(*)∈Gn_(n).Moreover,we also prove that if Xi is a Gateaux differentiability space,then there exist a real numberα>0 and a ball-covering(B)i of Xi such that(B)i isα-off the origin if and only if there exist a real numberα>0 and a ball-covering B of l^(∞)(X_(i))such that(B)isα-off the origin.展开更多
In this paper we show that the unit ball of an infinite dimensional commutative C-algebra lacks strongly exposed points, so they have no predual. Also in the second part, we use the concept of strongly exposed points ...In this paper we show that the unit ball of an infinite dimensional commutative C-algebra lacks strongly exposed points, so they have no predual. Also in the second part, we use the concept of strongly exposed points in the Frechet differentiability of support convex functions.展开更多
Three characteristics of the very rotund space ape proved and the relationships between the very rotund space and the geometrical properties of Banach space are discussed. Also connection between the weakly exposed po...Three characteristics of the very rotund space ape proved and the relationships between the very rotund space and the geometrical properties of Banach space are discussed. Also connection between the weakly exposed points and Radon Nikodym-property is established.展开更多
文摘We express the set of exposed points in terms of rotund points and non-smooth points.As long as we have Banach spaces each time＂bigger＂,we consider sets of non-smooth points each time＂smaller＂.
文摘In this paper we derive sufficient conditions for strict convexity of subsets in a complete simply connected smooth Riemanian manifold without focal points in terms of local and global exposed points.
文摘Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth points which are also non-exposed.
基金supported by the“China Natural Science Fund”under grant 11871181the“China Natural Science Fund”under grant 12026423.
文摘In this paper,we prove that(X,p)is separable if and only if there exists a w^(*)-lower semicontinuous norm sequence{p_(n)}_(n=1)^(∞)of(X^(*),p)such that(1)there exists a dense subset G_(n)of X^(*)such that p_(n)is Gateaux differentiable on G_(n)and dp_(n)(Gn_(n))■X for all n∈N;(2)p_(n)≤p and p_(n)→p uniformly on each bounded subset of X^(*);(3)for anyα∈(0,1),there exists a ball-covering{B(x^(*)i,n,Ti,n)}∞i=1 of(X^(*),p_(n))such that it isα-off the origin and x_(i,n)^(*)∈Gn_(n).Moreover,we also prove that if Xi is a Gateaux differentiability space,then there exist a real numberα>0 and a ball-covering(B)i of Xi such that(B)i isα-off the origin if and only if there exist a real numberα>0 and a ball-covering B of l^(∞)(X_(i))such that(B)isα-off the origin.
基金Supported by the Research Institute of Fundamental Sciences, Tabriz, Iran.
文摘In this paper we show that the unit ball of an infinite dimensional commutative C-algebra lacks strongly exposed points, so they have no predual. Also in the second part, we use the concept of strongly exposed points in the Frechet differentiability of support convex functions.
文摘Three characteristics of the very rotund space ape proved and the relationships between the very rotund space and the geometrical properties of Banach space are discussed. Also connection between the weakly exposed points and Radon Nikodym-property is established.
