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<!DOCTYPE html><html><head><title>Join Geometries: A Theory of Convex Sets and Linear Geometry - W. Prenowitz, J. Jantosciak - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/Join_Geometries.html?id=NFDvAAAAMAAJ"/><meta property="og:url" content="https://books.google.com/books/about/Join_Geometries.html?id=NFDvAAAAMAAJ"/><meta name="title" content="Join Geometries"/><meta name="description" content="The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical world and seem to be conceptually unrelated. However, a natural setting for their study is provided by the concept of convex set, which is compara tively new in the history of geometrical ideas. The familiarfigures can then appear as convex sets, boundaries of convex sets, or finite unions of convex sets. Moreover, two basic types of figure in linear geometry are special cases of convex set: linear space (point, line, and plane) and halfspace (ray, halfplane, and halfspace). Therefore we choose convex set to be the central type of figure in our treatment of geometry. How can the wealth of geometric knowledge be organized around this idea? By defini tion, a set is convex if it contains the segment joining each pair of its points; that is, if it is closed under the operation of joining two points to form a segment. 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Prenowitz</span></a>, <a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22J.+Jantosciak%22" class="secondary"><span dir=ltr>J. Jantosciak</span></a></div><div><span dir=ltr>Springer New York</span>, <span dir=ltr>16 Apr 1979</span> - <a class="secondary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22&source=gbs_ge_summary_r&cad=0"><span dir=ltr>Mathematics</span></a> - <span dir=ltr>534 pages</span></div></div><div id=synopsis><div id=synopsis-window><div id=synopsistext dir=ltr class="sa">The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. The postulational basis of the subject will be radically revised in order to construct a broad-scale and conceptually unified treatment. The familiar figures of classical geometry-points, segments, lines, planes, triangles, circles, and so on-stem from problems in the physical world and seem to be conceptually unrelated. However, a natural setting for their study is provided by the concept of convex set, which is compara tively new in the history of geometrical ideas. The familiarfigures can then appear as convex sets, boundaries of convex sets, or finite unions of convex sets. Moreover, two basic types of figure in linear geometry are special cases of convex set: linear space (point, line, and plane) and halfspace (ray, halfplane, and halfspace). Therefore we choose convex set to be the central type of figure in our treatment of geometry. How can the wealth of geometric knowledge be organized around this idea? By defini tion, a set is convex if it contains the segment joining each pair of its points; that is, if it is closed under the operation of joining two points to form a segment. But this is precisely the basic operation in Euclid.</div></div></div><div class="search_box_wrapper"></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="search_anchor"></a>From inside the book</h3><div id=search class=about_content><form action=/books?source=gbs_search_r&cad=1 id=search_form method=get> <input type=hidden name="redir_esc" value="y"><input type=hidden name="id" value="NFDvAAAAMAAJ"><input type=hidden name="dq" value="bibliogroup:"Undergraduate Texts in Mathematics""><input type="hidden" name="focus" value="searchwithinvolume"><input id=search_form_input type=text maxlength=1024 name=q value="bibliogroup:"Undergraduate Texts in Mathematics"" title="Search" accesskey="i" aria-label="Search in this book" style="width:160px"><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script> <input type=submit value="Search"></form><div id=search_v><div class="gb-page-no-results-element"><p class="message">Your search - <b>bibliogroup:"Undergraduate Texts in Mathematics"</b> - did not match any documents.</p><p class="try-all-volumes">Try this search over all volumes: <a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22" dir=ltr><b>bibliogroup:"Undergraduate Texts in Mathematics"</b></a></p></div><div id="searchwithinvolume" tabindex="0"><p align=left class="sitb-info">Results 1-0 of 0</p></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Abstract Theory of Join Operations </span></span></div></td><td class="toc_number" align=right>46</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap">51 </span></div></td><td class="toc_number" align=right>92</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Generation of Convex SetsConvex Hulls </span></span></div></td><td class="toc_number" align=right>124</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Copyright</span></span></div></td><td class="toc_number" align=right></td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/>11 other sections not shown</div><div style="clear:left; height:1em"></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN0387903402&id=NFDvAAAAMAAJ'>View all</a></h3><div id=book_other_versions class=about_content><div id=book_other_versions_v><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=OTTVBwAAQBAJ&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=OTTVBwAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=OTTVBwAAQBAJ&printsec=frontcover&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Join Geometries: A Theory of Convex Sets and Linear Geometry</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22W.+Prenowitz%22" class="secondary"><span dir=ltr>W. Prenowitz</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22J.+Jantosciak%22" class="secondary"><span dir=ltr>J. Jantosciak</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2012</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=vMTRngEACAAJ&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=vMTRngEACAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=vMTRngEACAAJ&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Join Geometries: A Theory of Convex Sets and Linear Geometry</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22W.+Prenowitz%22" class="secondary"><span dir=ltr>W. Prenowitz</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22J.+Jantosciak%22" class="secondary"><span dir=ltr>J. Jantosciak</span></a></span><br/><span><span style="color:#999">No preview available</span> - 2011</span><br/></span></div></td><td align=right></td></tr></table></div></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"Join Geometries","authors":"W. Prenowitz, J. Jantosciak","bib_key":"ISBN:9781461394389","pub_date":"6 Dec 2012","snippet":"The main object of this book is to reorient and revitalize classical geometry in a way that will bring it closer to the mainstream of contemporary mathematics. 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The ...","subject":"Mathematics","info_url":"https://books.google.com.sg/books?id=vMTRngEACAAJ\u0026dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=vMTRngEACAAJ\u0026dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/content?id=vMTRngEACAAJ\u0026printsec=frontcover\u0026img=1\u0026zoom=1","num_pages":0,"viewability":4,"preview":"noview","embeddable":false,"my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false,"subtitle":"A Theory of Convex Sets and Linear Geometry"}]);})();</script></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="word_cloud_anchor"></a>Common terms and phrases</h3><div id=word_cloud class=about_content><div id=word_cloud_v><style type="text/css">.cloud9 {color: #7777cc;font-size: 10px;}.cloud8 {color: #6963CC;font-size: 10.5px;}.cloud7 {color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=a(bc&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>a(bc</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=a%E2%82%81&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>a₁</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=a%E2%82%82&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>a₂</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=ab)c&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>ab)c</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=algebraic&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>algebraic</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=analogue&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>analogue</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=associative+law&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>associative law</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=b%E2%82%81&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>b₁</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=closed+halfspace&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>closed halfspace</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=closed+segment&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>closed segment</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=closure&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>closure</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=commutative+laws&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>commutative laws</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Compare+Corollary&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Compare Corollary</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Compare+Exercise&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Compare Exercise</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Compare+Theorem&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Compare Theorem</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=component&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>component</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=contains&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>contains</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=convex+hull&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>convex hull</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=convex+o-cone&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>convex o-cone</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=convex+set&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>convex set</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=defined&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=definition&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>definition</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=denoted&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>denoted</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=distinct+points&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>distinct points</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=elements&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>elements</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=end+of+Section&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>end of Section</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=endpoint&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>endpoint</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=equivalent&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>equivalent</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Euclidean+3-space&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Euclidean 3-space</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Euclidean+geometry&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Euclidean geometry</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Euclidean+join&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>Euclidean join</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Euclidean+plane&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Euclidean plane</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=example&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>example</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=exchange+geometry&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>exchange geometry</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=extension&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>extension</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=extreme+o-ray&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>extreme o-ray</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=extreme+point&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>extreme point</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=extreme+set&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>extreme set</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=face&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>face</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=factor+geometry&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>factor geometry</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=family+of+o-rays&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>family of o-rays</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=finite&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>finite</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Hence&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>Hence</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=implies&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>implies</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=interior+point&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>interior point</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=intersection&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>intersection</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=isomorphic&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>isomorphic</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=join+geometry&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>join geometry</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=join+of+points&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>join of points</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=join+operation&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>join operation</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=join+system&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>join system</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=L-cone&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>L-cone</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=L-ray&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>L-ray</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=linear+hull&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>linear hull</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=linear+space&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>linear space</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=linearly+independent&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>linearly independent</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=member+of+F&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>member of F</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=metric+interior&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>metric interior</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=noncollinear&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>noncollinear</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=nonempty+linear+set&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>nonempty linear set</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=notion&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>notion</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=open+set&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>open set</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=opposite+o-rays&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>opposite o-rays</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=ordered+geometry+let&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>ordered geometry let</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=P%E2%82%81&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>P₁</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=polytope&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>polytope</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=postulates&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>postulates</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=PROOF&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>PROOF</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Prove&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Prove</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=real+numbers&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>real numbers</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=relation&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>relation</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=result&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>result</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=S%E2%82%81&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>S₁</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=satisfies&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>satisfies</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=set+of+points&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>set of points</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Similarly&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Similarly</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=supporting+hyperplane&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>supporting hyperplane</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Suppose&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Suppose</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=theory&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>theory</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=triode&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>triode</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=union&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>union</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=Verify&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Verify</span></a> <a href="https://books.google.com.sg/books?id=NFDvAAAAMAAJ&q=vertices&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>vertices</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>Join Geometries: A Theory of Convex Sets and Linear Geometry</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Applied+Mathematical+Sciences%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Applied Mathematical Sciences</span></i></a><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Volume 11 of Undergraduate Texts in Mathematics</span></i></a>, <span dir=ltr>ISSN</span> <span dir=ltr>0172-6056</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Authors</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22W.+Prenowitz%22&source=gbs_metadata_r&cad=5"><span dir=ltr>W. Prenowitz</span></a>, <a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22J.+Jantosciak%22&source=gbs_metadata_r&cad=5"><span dir=ltr>J. Jantosciak</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>illustrated</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Springer New York, 1979</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Original from</span></td><td class="metadata_value"><span dir=ltr>the University of Michigan</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Digitized</span></td><td class="metadata_value"><span dir=ltr>5 Feb 2010</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0387903402, 9780387903408</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>534 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry%22" itemprop="url" dir=ltr><span itemprop="title">Geometry</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry+General%22" itemprop="url" dir=ltr><span itemprop="title">General</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+Algebraic%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Geometry / Algebraic</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+Analytic%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Geometry / Analytic</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+General%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Geometry / General</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/Join_Geometries.bibtex?id=NFDvAAAAMAAJ&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Join_Geometries.enw?id=NFDvAAAAMAAJ&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Join_Geometries.ris?id=NFDvAAAAMAAJ&output=ris"><span dir=ltr>RefMan</span></a></td></tr></table></div><div style="clear:both"></div></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBrowsingHistoryEnabled:1, IsZipitFolderCollectionEnabled:1, IsBooksUnifiedLeftNavEnabled:1, IsBooksRentalEnabled:1});_OC_Run({"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3DNFDvAAAAMAAJ%26q%3Dbibliogroup:%2522Undergraduate%2BTexts%2Bin%2BMathematics%2522%26dq%3Dbibliogroup:%2522Undergraduate%2BTexts%2Bin%2BMathematics%2522%26hl%3Den%26sa%3DX%26redir_esc%3Dy\u0026hl=en"}, {"volume_id":"NFDvAAAAMAAJ","is_ebook":false,"volumeresult":{"has_flowing_text":false,"has_scanned_text":false,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"sample_url":"https://play.google.com/books/reader?id=NFDvAAAAMAAJ\u0026source=gbs_atb_hover","is_browsable":false,"is_public_domain":false}, {});</script><div id="footer_table" style="font-size:83%;text-align:center;position:relative;top:20px;height:4.5em;margin-top:2em"><div style="margin-bottom:8px"><a href="/intl/en/googlebooks/about.html"><nobr><nobr>About Google Books</nobr></nobr></a> - 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