MATHEMATICA BOHEMICA, Vol. 128, No. 2, pp. 169-178 (2003)

A note on equality of functional envelopes

Martin Kruzik

Martin Kruzik, Center of Advanced European Studies and Research, Friedensplatz 16, 53 111 Bonn, Germany (address for correspondence), e-mail: and Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 4, 182 08 Praha 8, Czech Republic, e-mail:

Abstract: We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in $\R^{m\times n}$, $\min(m,n)\le2$, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

Keywords: extreme points, polyconvexity, quasiconvexity, rank-1 convexity

Classification (MSC2000): 49J45

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number] [Journals Homepage]
© 2004–2010 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition