Based　on　the　＂convexly　separable＂　concept,　we　present　a　solid　geometric　theory　and　a　new　general　framework　to　design　piecewise　linear　classifiers　for　two　arbitrarily　complicated　nonintersecting　classes　by　using　a　＂multiconlitron,＂　which　is　a　union　of　multiple　conlitrons　that　comprise　a　set　of　hyperplanes　or　linear　functions　surrounding　a　convex　region　for　separating　two　convexly　separable　datasets.　We　propose　a　new　iterative　algorithm　called　the　cross　distance　minimization　algorithm　(CDMA)　to　compute　hard　margin　non-kernel　support　vector　machines　(SVMs)　via　the　nearest　point　pair　between　two　convex　polytopes.　Using　CDMA,　we　derive　two　new　algorithms,　i.e.,　the　support　conlitron　algorithm　(SCA)　and　the　support　multiconlitron　algorithm　(SMA)　to　construct　support　conlitrons　and　support　multiconlitrons,　respectively,　which　are　unique　and　can　separate　two　classes　by　a　maximum　margin　as　in　an　SVM.　Comparative　experiments　show　that　SMA　can　outperform　linear　SVM　on　many　of　the　selected　databases　and　provide　similar　results　to　radial　basis　function　SVM　on　some　of　them,　while　SCA　performs　better　than　linear　SVM　on　three　out　of　four　applicable　databases.　Other　experiments　show　that　SMA　and　SCA　may　be　further　improved　to　draw　more　potential　in　the　new　research　direction　of　piecewise　linear　learning.