This is an announcement for the paper "On random intersections of two convex bodies. Appendix to: "Isoperimetry of waists and local versus global asymptotic convex geometries" by R.Vershynin" by Mark Rudelson and Roman Vershynin. Abstract: In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections). Archive classification: Functional Analysis Mathematics Subject Classification: 52A20, 46B07 The source file(s), localglobal-appendix.tex: 8336 bytes, is(are) stored in gzipped form as 0404502.gz with size 3kb. The corresponding postcript file has gzipped size 24kb. Submitted from: vershynin@math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0404502 or http://arXiv.org/abs/math.FA/0404502 or by email in unzipped form by transmitting an empty message with subject line uget 0404502 or in gzipped form by using subject line get 0404502 to: math@arXiv.org.