This is an announcement for the paper "A note on lower bounds of martingale measure densities" by D. Rokhlin and W. Schachermayer. Abstract: For a given element $f\in L^1$ and a convex cone $C\subset L^\infty$, $C\cap L^\infty_+=\{0\}$ we give necessary and sufficient conditions for the existence of an element $g\ge f$ lying in the polar of $C$. This polar is taken in $(L^\infty)^*$ and in $L^1$. In the context of mathematical finance the main result concerns the existence of martingale measures, whose densities are bounded from below by prescribed random variable. Archive classification: Functional Analysis Mathematics Subject Classification: 46E30 Remarks: 9 pages The source file(s), SCH_P4.TEX: 22410 bytes, is(are) stored in gzipped form as 0505411.gz with size 8kb. The corresponding postcript file has gzipped size 46kb. Submitted from: rokhlin@math.rsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0505411 or http://arXiv.org/abs/math.FA/0505411 or by email in unzipped form by transmitting an empty message with subject line uget 0505411 or in gzipped form by using subject line get 0505411 to: math@arXiv.org.