Before revision:  In summary, TA and the\textit{  Spherical-CHM provide a convenient geometric setting for efficient solution to large-scale CHM and related problems, such as computing all vertices of conv(S)\emph{ }\hspace{0pt}%DIFAUXCMD , called }\emph{ }\hspace{0pt}%DIFAUXCMD } .
After revision:  In summary, TA and the\textit{ Spherical-CHM provide a convenient geometric setting for efficient solution to large-scale CHM and related problems, such as computing all vertices of conv(S)\emph{ }\hspace{0pt}%DIFAUXCMD , called }\emph{ for solving exact or approximate irredundancy problem. Computationally, we have considered CHM, LP and Strict LP Feasibility and the Irredundancy problem. Based on substantial amount of computing, Spherical-TA achieves better efficiency than state of the art algorithms. Leveraging on the efficiency of Spherical-TA, we propose AVTA^+ as a pre-processing step for data reduction which arises in such applications as in computing the Minimum Volume Enclosing Ellipsoid \mbox{%DIFAUXCMD \cite{moshtagh2005minimum}\hspace{0pt}%DIFAUXCMD } .