{ localUrl: '../page/convex_set.html', arbitalUrl: 'https://arbital.com/p/convex_set', rawJsonUrl: '../raw/5bs.json', likeableId: '3051', likeableType: 'page', myLikeValue: '0', likeCount: '3', dislikeCount: '0', likeScore: '3', individualLikes: [ 'EricBruylant', 'NateSoares', 'EricRogstad' ], pageId: 'convex_set', edit: '7', editSummary: '', prevEdit: '6', currentEdit: '7', wasPublished: 'true', type: 'wiki', title: 'Convex set', clickbait: 'A set that contains all line segments between points in the set', textLength: '1192', alias: 'convex_set', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JessicaTaylor', editCreatedAt: '2016-07-13 04:40:03', pageCreatorId: 'JessicaTaylor', pageCreatedAt: '2016-07-13 04:05:05', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '1', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '51', text: 'A convex set is a [-3jz] of [3w0 vectors] that contains all line segments between vectors in the set. Consider the following shape:\n\n![a convex set](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Convex_polygon_illustration1.svg/634px-Convex_polygon_illustration1.svg.png)\n\nAs shown, a line segment between two points $x$ and $y$ in this shape lies entirely within the shape. In fact, this is true for any pair of points in the shape. Therefore, this shape is convex. For comparison, the following shape is not convex:\n\n![a non-convex set](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Convex_polygon_illustration2.svg/634px-Convex_polygon_illustration2.svg.png)\n\nThe fact that part of the line segment between $x$ and $y$ (both inside the shape) lies outside the shape proves that this shape is not convex.\n\n\nFormally, a set $S$ is convex if\n\n$$\\forall x, y \\in S, \\theta \\in [0, 1]: \\theta x + (1 - \\theta) y \\in S$$\n\n(images are from Wikipedia: [here](https://en.wikipedia.org/wiki/File:Convex_polygon_illustration1.svg) and [here](https://en.wikipedia.org/wiki/File:Convex_polygon_illustration2.svg))\n\n[todo: use non-Wikipedia images and remove the attribution]', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JessicaTaylor' ], childIds: [], parentIds: [ 'set_mathematics', 'convex' ], commentIds: [], questionIds: [], tagIds: [ 'start_meta_tag', 'needs_summary_meta_tag', 'split_by_mastery_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18896', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-08-20 11:56:58', auxPageId: 'start_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18895', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-08-20 11:56:52', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18893', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-08-20 11:56:50', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18891', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-08-20 11:56:48', auxPageId: 'split_by_mastery_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18890', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-08-20 11:56:26', auxPageId: 'convex', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18888', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-08-20 11:56:18', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18886', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-08-20 11:56:17', auxPageId: 'set_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16679', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-07-14 00:31:17', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16677', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-14 00:31:15', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16643', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-13 20:45:08', auxPageId: 'needs_summary_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16642', pageId: 'convex_set', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-13 20:43:37', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16621', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '7', type: 'newEdit', createdAt: '2016-07-13 04:40:03', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16620', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '6', type: 'newEdit', createdAt: '2016-07-13 04:38:39', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16619', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '5', type: 'newEdit', createdAt: '2016-07-13 04:37:24', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16618', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '4', type: 'newEdit', createdAt: '2016-07-13 04:15:09', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16617', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '3', type: 'newEdit', createdAt: '2016-07-13 04:14:46', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16616', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'newTag', createdAt: '2016-07-13 04:12:38', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16615', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'deleteTag', createdAt: '2016-07-13 04:12:28', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16613', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'deleteTag', createdAt: '2016-07-13 04:12:26', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16611', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '2', type: 'newEdit', createdAt: '2016-07-13 04:12:11', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16608', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'newTag', createdAt: '2016-07-13 04:05:07', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16609', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'newTag', createdAt: '2016-07-13 04:05:07', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16610', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '0', type: 'newTag', createdAt: '2016-07-13 04:05:07', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16607', pageId: 'convex_set', userId: 'JessicaTaylor', edit: '1', type: 'newEdit', createdAt: '2016-07-13 04:05:05', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: { moreTechnical: { likeableId: '3495', likeableType: 'contentRequest', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '77', pageId: 'convex_set', requestType: 'moreTechnical', createdAt: '2016-09-06 04:09:03' } } }