84:
74:
53:
170:
163:
22:
184:
I or III (light green in the picture), meaning the join of both endpoints equals one of them, and likewise for the meet. There are two exceptions, viz. the line (2,5) to (6,3), and the line (3,6) to (6,3), for which the picture shows (in light red) the rectangles for join and meet construction; both
260:
I see your point. I didn't dive deep enough into
Felsner.Knauer.2019 to understand theorem 4; I had just searched the keyword "convex". I don't have a good general definition of polytope either. (If I were to come up with one, I'd try solutions sets of conjunctions of disjunctions of linear
261:
inequations. This would probably admit too many degenerate forms as polytopes, but possibly many of them would be ruled-out by the distributivity condition.) I uploaded a picture of the convex hull of the above polygon; it might serve as an illustration to the article. -
233:
Do you have a good, standard, well-accepted definition of polyhedron/polytope that allows nonconvexity and applies in arbitrary dimensions? I don't. Even in three dimensions the definitional issue is a mess; see
195:
If I'm right, the restriction to convex polytops should be removed from the article (in fact, I didn't find it in
Felsner.Knauer.2011), and the picture might serve to illustrate a simple 2-dimentional example. -
177:
I believe the polygon (1,1) / (2,5) / (3,6) / (7,7) / (6,3) / (2,2) / (1,1) satisfies the definition from
Felsner.Knauer.2019 (see below); however, as can be seen in the picture, it is not convex.
140:
180:
All connecting lines between any two vertices are visible in the picture (interior connections in light grey), most of them approach their endpoints within the
294:
130:
289:
106:
192:, by some clever estimation arguments, the same can be shown for all points of the perimeter, and finally for all points of the polygon area.
97:
58:
33:
211:
Being a better hacker than mathematician, I ran a computer program to check all possibilities, with a resolution of 1
185:
joins and both meets are within the polygon. Thus, at least joins and meets of all vertices are within the polygon.
266:
224:
201:
243:
21:
262:
220:
197:
39:
83:
239:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
73:
52:
169:
283:
162:
270:
247:
228:
205:
102:
235:
79:
238:. And Theorem 4 of Felsner et al appears to explicitly assume convexity. —
181:
219:, and didn't find a violation of the Felsner.Knauer.2019 definition. -
168:
161:
15:
101:, a collaborative effort to improve the coverage of
8:
47:
19:
49:
7:
95:This article is within the scope of
38:It is of interest to the following
14:
295:Low-priority mathematics articles
115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
82:
72:
51:
20:
290:Stub-Class mathematics articles
135:This article has been rated as
1:
271:10:16, 15 November 2019 (UTC)
248:19:16, 14 November 2019 (UTC)
229:18:55, 14 November 2019 (UTC)
206:12:44, 14 November 2019 (UTC)
109:and see a list of open tasks.
311:
134:
67:
46:
141:project's priority scale
98:WikiProject Mathematics
174:
166:
28:This article is rated
236:Polyhedron#Definition
172:
165:
121:mathematics articles
175:
167:
90:Mathematics portal
34:content assessment
155:
154:
151:
150:
147:
146:
302:
263:Jochen Burghardt
221:Jochen Burghardt
198:Jochen Burghardt
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
310:
309:
305:
304:
303:
301:
300:
299:
280:
279:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
308:
306:
298:
297:
292:
282:
281:
278:
277:
276:
275:
274:
273:
253:
252:
251:
250:
240:David Eppstein
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
307:
296:
293:
291:
288:
287:
285:
272:
268:
264:
259:
258:
257:
256:
255:
254:
249:
245:
241:
237:
232:
231:
230:
226:
222:
218:
214:
210:
209:
208:
207:
203:
199:
193:
191:
186:
183:
178:
171:
164:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
216:
212:
194:
189:
187:
179:
176:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
213:length unit
173:Convex hull
112:Mathematics
103:mathematics
59:Mathematics
284:Categories
30:Stub-class
158:Convexity
182:quadrant
139:on the
217:pixels
215:≅ 100
36:scale.
190:guess
267:talk
244:talk
225:talk
202:talk
131:Low
286::
269:)
246:)
227:)
204:)
188:I
265:(
242:(
223:(
200:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.