Let K be a(bounded)closed uniformly convex subset of a Banach space X.We show that(i)the nearest point map is well-defined and always continuous from X onto K,(ii)there is a reflexive space Y with a uniform rotund in ...Let K be a(bounded)closed uniformly convex subset of a Banach space X.We show that(i)the nearest point map is well-defined and always continuous from X onto K,(ii)there is a reflexive space Y with a uniform rotund in every direction norm such that Y contains K as a subset and the nearest point map PK:Y→K is uniformly continuous from any bounded set containing K onto K.展开更多
Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the ...Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the metric space(B(X),H).For x∈X and F∈B_(G)(X),we denote the nearest point problem inf{||x-g||:g∈G}by min(x,G)and the mutually nearest point problem inf{||f-g||:f∈ F,g∈G}by min(F,G).In this paper,parallel to well-posedness of the problems min(a:,G)and mm(F,G)which are defined by De Blasi et al.,we further introduce the weak well-posedness of the problems min(x,G)and min(F,G).Under the assumption that the Banach space X has some geometric properties,we prove a series of results on weak well-posedness of min(x,G)and min(F,G).We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.展开更多
In this study, we extend our previous adaptive steganographic algorithm to support point geometry. For the purpose of the vertex decimation process presented in the previous work, the neighboring information between p...In this study, we extend our previous adaptive steganographic algorithm to support point geometry. For the purpose of the vertex decimation process presented in the previous work, the neighboring information between points is necessary. Therefore, a nearest neighbors search scheme, considering the local complexity of the processing point, is used to determinate the neighbors for each point in a point geometry. With the constructed virtual connectivity, the secret message can be embedded successfully after the vertex decimation and data embedding processes. The experimental results show that the proposed algorithm can preserve the advantages of previous work, including higher estimation accuracy, high embedding capacity, acceptable model distortion, and robustness against similarity transformation attacks. Most importantly, this work is the first 3D steganographic algorithm for point geometry with adaptation.展开更多
基金Supported by NSFC(Grant Nos.12071389,12471132)the Natural Science Foundation of Fujian Province(Grant No.2024J01028)+1 种基金Fujian Province Science Foundation for Youths(Grant No.2022J05279)Xiamen University of Technology high-level talents research launch project(Grant No.YKJ22012R)。
文摘Let K be a(bounded)closed uniformly convex subset of a Banach space X.We show that(i)the nearest point map is well-defined and always continuous from X onto K,(ii)there is a reflexive space Y with a uniform rotund in every direction norm such that Y contains K as a subset and the nearest point map PK:Y→K is uniformly continuous from any bounded set containing K onto K.
基金Supported by the NSFC(Grant No.11671252)the NSFC(Grant No.11771278)。
文摘Let G be a nonempty closed subset of a Banach space X.Let B(X)be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and B_(G)(X)={A∈B(X):A∩G=φ},where the closure is taken in the metric space(B(X),H).For x∈X and F∈B_(G)(X),we denote the nearest point problem inf{||x-g||:g∈G}by min(x,G)and the mutually nearest point problem inf{||f-g||:f∈ F,g∈G}by min(F,G).In this paper,parallel to well-posedness of the problems min(a:,G)and mm(F,G)which are defined by De Blasi et al.,we further introduce the weak well-posedness of the problems min(x,G)and min(F,G).Under the assumption that the Banach space X has some geometric properties,we prove a series of results on weak well-posedness of min(x,G)and min(F,G).We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.
基金supported by the National Science Council under Grant No. NSC98-2221-E-468-017 and NSC 100-2221-E-468-023the Research Project of Asia University under Grant No. 100-A-04
文摘In this study, we extend our previous adaptive steganographic algorithm to support point geometry. For the purpose of the vertex decimation process presented in the previous work, the neighboring information between points is necessary. Therefore, a nearest neighbors search scheme, considering the local complexity of the processing point, is used to determinate the neighbors for each point in a point geometry. With the constructed virtual connectivity, the secret message can be embedded successfully after the vertex decimation and data embedding processes. The experimental results show that the proposed algorithm can preserve the advantages of previous work, including higher estimation accuracy, high embedding capacity, acceptable model distortion, and robustness against similarity transformation attacks. Most importantly, this work is the first 3D steganographic algorithm for point geometry with adaptation.
