Spatial Trimming, with Applications to Robustify Sample Spatial Quantile and Outlyingness Functions, and to Construct a New Robust Scatter Estimator Satyaki Mazumder 1 and Robert Serfling 2 University of Texas at Dallas April, 2010 1 Department of Mathematical Sciences, University of Texas at Dallas, Richard- son, Texas 75083-0688, USA. 2 Department of Mathematical Sciences, University of Texas at Dallas, Richard- son, Texas 75083-0688, USA. Email: serfling@utdallas.edu. Website: www.utdallas.edu/∼serfling. Support by NSF Grants DMS-0103698 and DMS- 0805786 and NSA Grant H98230-08-1-0106 is gratefully acknowledged.