## 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.

### MSC:

 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

### Keywords:

semicontinuity; rough convexity; boundedness; rank 1 convexity
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### References:

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