TY - JOUR

T1 - The Strong Continuity of Convex Functions

AU - Malozemov, V. N.

AU - Plotkin, A. V.

AU - Tamasyan, G. Sh.

N1 - Malozemov, V.N., Plotkin, A.V. & Tamasyan, G.S. The Strong Continuity of Convex Functions. Vestnik St.Petersb. Univ.Math. 51, 244–248 (2018). https://doi.org/10.3103/S1063454118030056

PY - 2018/9/4

Y1 - 2018/9/4

N2 - A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.

AB - A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.

KW - constant of strong continuity

KW - convex function

KW - strong continuity

UR - http://www.scopus.com/inward/record.url?scp=85052839845&partnerID=8YFLogxK

U2 - 10.3103/S1063454118030056

DO - 10.3103/S1063454118030056

M3 - Article

VL - 51

SP - 244

EP - 248

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -