Local boundedness and continuity of generalized convex functions. (English) Zbl 0815.26004

Summary: This article deals with generalizations of the usual convexity of real- valued functions in such a manner the “convex” is extended to “\({\mathcal F}\)-convex”, and \({\mathcal F}\)-convexity is required only on straight lines with directions from a given cone \(K\). Under certain assumptions on the generating family \(\mathcal F\) and on \(K\), for functions of such kind (called \({\mathcal F}\)-convex on \(K\)-lines) local boundedness and continuity properties are obtained. The main results are applied to a number of examples. In particular, Morrey’s rank 1 convexity and a special type of “rough convexity” are considered.


26B25 Convexity of real functions of several variables, generalizations
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A51 Convexity of real functions in one variable, generalizations
52A41 Convex functions and convex programs in convex geometry
Full Text: DOI


[1] DOI: 10.1007/BF00279992 · Zbl 0368.73040
[2] DOI: 10.1090/S0002-9904-1937-06549-9 · Zbl 0016.35202
[3] Beckenbach E.F., Trans. Am. Math. SOC 58 pp 220– (1945)
[4] Beckenbach E.F., Pac. Journ. Math 3 pp 291– (1953) · Zbl 0050.10105
[5] DOI: 10.1017/S1446788700018644
[6] DOI: 10.1007/BF00932539 · Zbl 0325.26007
[7] Berge C., Espaces Topologiques (1966)
[8] DOI: 10.1080/02331938708843207 · Zbl 0609.52001
[9] DeFinetti B., Ann. Math. Pura Appl 30 pp 102– (1949)
[10] Dacorogna B., Applied Mathematical Sciences 78 (1989)
[11] Dolecki S., SIAM Journ. Control Opt 16 pp 227– (1978) · Zbl 0397.46013
[12] Favard J., Problèmes d’extremum relatifs aux courves convexes I, Ann. Ecole Norm. Sup 46 pp 357– (1929)
[13] DOI: 10.1080/02331938308842832 · Zbl 0514.26003
[14] DOI: 10.1080/02331938508843063 · Zbl 0585.26008
[15] DOI: 10.1137/0327055 · Zbl 0686.52006
[16] März, C. 1990. ”Ein Optimierungsproblem zu Favards ”fonction penetrante”. Universität Leipzig. Diplomarbeit
[17] Morrey C.B., Pac.Journ. Math 2 pp 25– (1952) · Zbl 0046.10803
[18] Morrey C.B., Multiple Integrals in the Calculus of Variations (1966) · Zbl 0142.38701
[19] Phu H. X. Univ. Augsburg 1990 Gamma-subdifferential and gamma-convexity of functions on the real line, Inst. für Math., Report No. 259
[20] Phu H. X. Univ. Augsburg 1991 Gamma-subdifferential and gamma-convexity of functions on a Banach space, Inst. für Math. Report No. 270
[21] Roberts A.W., Convex Functions (1973) · Zbl 0271.26009
[22] Scarf H. The optimality of (s,S) -policies in the dynamic inventory problem 1959 Proc. of the First Stanford Symp · Zbl 0203.22102
[23] Schaible S., Generalized Concavity in Optimization and Economics (1981)
[24] Söllner, B. 1991. ”Eigenschafteny-grobkonvexer Mengen und Funktionen. Untersuchungen zu einer neuen Verallgemeinerung der Konvexitat”. Universitat Leipzig. Diplomarbeit
[25] DOI: 10.1007/BF00932614 · Zbl 0268.90057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.