×

The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity. (English) Zbl 1152.26012

Given a function \(f:\mathbb{R}^{n}\rightarrow[-\infty,\infty]\), its “perspective” is the function \(\breve{f}\) defined on \(\mathbb{R}^{n}\times\mathbb{R}_{+}^{\ast}\) by \(\breve {f}(x,y)=yf(\frac{x}{y})\). It is known that \(f\) is convex if and only if \(\breve{f}\) is convex. The article under review presents a nice application of this notion by providing a new, simple proof of the following result: if \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}\) is separately convex and positively homogeneous of degree 1, then it is convex. Counterexamples are provided to show that the theorem does not extend to higher dimensions, or to the case where \(f\) may take on infinite values. Finally, a notion of “generalized perspective” is introduced and used to study the convexity of a family of functions defined on the space of \(n\times n\) matrices.

MSC:

26B25 Convexity of real functions of several variables, generalizations
26A51 Convexity of real functions in one variable, generalizations
PDFBibTeX XMLCite
Full Text: Link