# The Strong Continuity of Convex Functions

Research output

### Abstract

A convex function deﬁned on an open convex set is known to be continuous at every point of this set. In actuality, a convex function has a strengthened continuity property. In this paper, we introduce the notion of strong continuity and demonstrate that a convex function possesses this property. The proof is based only on the deﬁnition of convexity and the Jensen’s inequality. A distinct constant (constant of strong continuity) is included in the deﬁnition of strong continuity. In the article, we give an unimprovable value for this constant in the case of convex functions. The constant of strong continuity depends, in particular, on the form of the norm introduced in the space of the arguments of a convex function. Polyhedral norm is of particular interest. With its use the constant of strong continuity can be easily calculated. This requires a ﬁnite number of values of the convex function.
Original language English 244-248 5 Vestnik St. Petersburg University: Mathematics 51 3 https://doi.org/10.3103/S1063454118030056 Published - 2018

### Cite this

@article{dc27f7e4e2374826a19942e09a85b4e9,
title = "The Strong Continuity of Convex Functions",
abstract = "A convex function deﬁned on an open convex set is known to be continuous at every point of this set. In actuality, a convex function has a strengthened continuity property. In this paper, we introduce the notion of strong continuity and demonstrate that a convex function possesses this property. The proof is based only on the deﬁnition of convexity and the Jensen’s inequality. A distinct constant (constant of strong continuity) is included in the deﬁnition of strong continuity. In the article, we give an unimprovable value for this constant in the case of convex functions. The constant of strong continuity depends, in particular, on the form of the norm introduced in the space of the arguments of a convex function. Polyhedral norm is of particular interest. With its use the constant of strong continuity can be easily calculated. This requires a ﬁnite number of values of the convex function.",
author = "Малоземов, {Василий Николаевич} and Тамасян, {Григорий Шаликович} and Плоткин, {Артем Владимирович}",
year = "2018",
doi = "10.3103/S1063454118030056",
language = "English",
volume = "51",
pages = "244--248",
journal = "Vestnik St. Petersburg University: Mathematics",
issn = "1063-4541",
number = "3",

}

In: Vestnik St. Petersburg University: Mathematics, Vol. 51, No. 3, 2018, p. 244-248.

Research output

TY - JOUR

T1 - The Strong Continuity of Convex Functions

AU - Малоземов, Василий Николаевич

AU - Тамасян, Григорий Шаликович

AU - Плоткин, Артем Владимирович

PY - 2018

Y1 - 2018

N2 - A convex function deﬁned on an open convex set is known to be continuous at every point of this set. In actuality, a convex function has a strengthened continuity property. In this paper, we introduce the notion of strong continuity and demonstrate that a convex function possesses this property. The proof is based only on the deﬁnition of convexity and the Jensen’s inequality. A distinct constant (constant of strong continuity) is included in the deﬁnition of strong continuity. In the article, we give an unimprovable value for this constant in the case of convex functions. The constant of strong continuity depends, in particular, on the form of the norm introduced in the space of the arguments of a convex function. Polyhedral norm is of particular interest. With its use the constant of strong continuity can be easily calculated. This requires a ﬁnite number of values of the convex function.

AB - A convex function deﬁned on an open convex set is known to be continuous at every point of this set. In actuality, a convex function has a strengthened continuity property. In this paper, we introduce the notion of strong continuity and demonstrate that a convex function possesses this property. The proof is based only on the deﬁnition of convexity and the Jensen’s inequality. A distinct constant (constant of strong continuity) is included in the deﬁnition of strong continuity. In the article, we give an unimprovable value for this constant in the case of convex functions. The constant of strong continuity depends, in particular, on the form of the norm introduced in the space of the arguments of a convex function. Polyhedral norm is of particular interest. With its use the constant of strong continuity can be easily calculated. This requires a ﬁnite number of values of the convex function.

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 -