A fundamental problem in linear programming, machine learning, and computational geometry is the%DIFDELCMD < {\it %%%  Convex Hull Membership (CHM)  : Given a point p and a subset S of n points in R^m, is p \in conv(S)?  The {\it Triangle Algorithm} (TA) computes p' \in conv(S) so that, either%DIFDELCMD < \Vert %%% p'- p%DIFDELCMD < \Vert %%% \leq \varepsilon R, R= \max \{%DIFDELCMD < \Vert %%% p -v%DIFDELCMD < \Vert%%% : v\in S\ {\it witness , i. e. the orthogonal bisector of pp' separates p from conv(S). By the%DIFDELCMD < {\it %%% Spherical}-CHM we mean a CHM, where p=0 ,%DIFDELCMD < \Vert %%% v%DIFDELCMD < \Vert%%% =1, \forall v \in S . First, we prove the equivalence of exact and approximate versions of CHM and Spherical-CHM. On the one hand, this makes it possible to state a simple O(1/\varepsilon^2) iteration TA, each taking O(n+m) time . On the other hand, using this iteration complexity we prove if for each p' \in conv(S) with%DIFDELCMD < \Vert %%% p%DIFDELCMD < \Vert %%% > \varepsilon that is not a witness there is v \in S with%DIFDELCMD < \Vert %%% p' - v%DIFDELCMD < \Vert %%% \geq \sqrt{1+ \varepsilon to O(1/\varepsilon). This matches complexity of Nesterov's fast-gradient method. The analysis also suggests a strategy for when the property does not hold at an iterate. Lastly, as an application of TA, we show how to solve strict LP feasibility as a dual of CHM. 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 } .
%DIFDELCMD < {\it %%% The it Convex Hull Membership (CHM) problem is : Given a point p and a subset S of n points in R^m, is p \in conv(S)? CHM is not only a fundamental problem in Linear Programming, Computational Geometry, Machine Learning and Statistics, it also serves as a query problem in many applications e.g. Topic Modeling, LP Feasibility, Data Reduction. The {\it Triangle Algorithm} (TA) %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert%%% \mbox{%DIFAUXCMD kalantari2015characterization {\it %DIFDELCMD < {\it %%% Spherical}-CHM is a CHM, where p=0 %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert%%% and each point in S has unit norm . First, we prove the equivalence of exact and approximate versions of CHM and Spherical-CHM. On the one hand, this makes it possible to state a simple version of the original TA . On the other hand, %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert %%% %DIFDELCMD < \Vert %%% we prove that under the satisfiability of a simple condition in each iteration, the complexity improves to O(1/\varepsilon). The analysis also suggests a strategy for when the property does not hold at an iterate. This suggests the\textit{Spherical-TA which first converts a given CHM into a Spherical-CHM before applying the algorithm. Next we introduce a series of applications of Spherical-TA. In particular, Spherical-TA serves as a fast version of vanilla TA to boost its efficiency. As an example, this results in a fast version of\emph{AVTA \mbox{%DIFAUXCMD \cite{awasthi2018robust}\hspace{0pt}%DIFAUXCMD , called }\emph{AVTA^+ 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 } .
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CHM is not only a fundamental problem in Linear Programming, Computational Geometry, Machine Learning and Statistics, it also serves as a query problem in many applications e.g. Topic Modeling, LP Feasibility, Data Reduction.
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computes p' \in conv(S) so that, either
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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