81:
71:
53:
22:
246:
Example 3 in the article shows that a circle, while not a convex set, is a convex metric space. I see you are suggesting another such example above. Feel free to add it in. One could also put in a blurb in the intro saying that convex metric spaces are not necessarily convex sets.
278:
What do you mean by a line with a closed interval removed? If you restrict the distance function on line to this subset, then it is not convex, and so how does this give contradiction to claim that convexity implies geodesic convexity?
155:
I think that we need to further stress the difference between, metric convexity and usual convexity, in that some sets that are non-convex in the usual sense are convex by this article's definition. For example, let
133:
265:
as you suggest. However, my point goes even further. Not only does metric convexity not imply usual convexity (as in the example of the circle), it does not imply
437:
127:
361:
geodesic metric = any two points can be joined by minimizing geodesic (that is also convex metric space in old german way, the definition here is wrong)
432:
380:
The first two are only equivalent for complete metric spaces. The second two are only equivalent for complete and locally compact metric spaces; see
103:
94:
58:
284:
33:
354:
lenght metric = distance is infimim of lengths of paths connecting two points (that is what called
Intrinsic metric in
300:
252:
21:
397:) of a non-locally compact complete intrinsic metric space in which two points are not joined by a geodesic.
313:
Convex metric space is old name geodesic space. In the article a wrong def is given. Why revert my edit...--
280:
409:
333:
39:
80:
328:
Your edit was reverted because the definitions given in the two articles define different concepts.
296:
248:
102:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
394:
390:
The plane with a bounded open line segment removed are a length metric space which is not convex.
381:
266:
86:
387:
The rational numbers with the induced
Euclidean metric are convex but not a length metric space.
70:
52:
403:
329:
270:
237:
355:
348:
370:
318:
269:(e.g. the line with a closed interval removed, or the plane with a closed disc removed).
426:
99:
343:
There are three standard defs which reflect different level of the same thing:
417:
374:
366:
337:
322:
314:
304:
288:
273:
256:
240:
76:
15:
347:
intrinsic metric= existance of almost midpoint (the def in
224:) / 2; otherwise, assume without loss of generality that
393:
There is an example due to C.J. Atkin (referenced in
98:, a collaborative effort to improve the coverage of
132:This article has not yet received a rating on the
212:lie on the same side of the "hole" , simply take
295:Thanks for the clarification in the article.
8:
176:the usual Euclidean distance inherited from
19:
47:
365:So all this should be in one place... --
49:
184:is convex, since given any two points
438:Unknown-priority mathematics articles
7:
92:This article is within the scope of
38:It is of interest to the following
14:
112:Knowledge:WikiProject Mathematics
433:Start-Class mathematics articles
115:Template:WikiProject Mathematics
79:
69:
51:
20:
418:12:36, 15 September 2008 (UTC)
384:. There are counterexamples:
375:12:37, 12 September 2008 (UTC)
1:
338:23:16, 9 September 2008 (UTC)
323:15:03, 9 September 2008 (UTC)
106:and see a list of open tasks.
289:23:09, 5 December 2017 (UTC)
305:05:14, 11 August 2007 (UTC)
274:12:25, 10 August 2007 (UTC)
454:
257:16:23, 9 August 2007 (UTC)
241:10:04, 9 August 2007 (UTC)
151:Metric and usual convexity
204:that is between them: if
131:
64:
46:
134:project's priority scale
95:WikiProject Mathematics
164:) be the metric space
28:This article is rated
118:mathematics articles
395:Hopf-Rinow theorem
382:Hopf-Rinow theorem
281:Lost-n-translation
267:geodesic convexity
261:I will add such a
87:Mathematics portal
34:content assessment
228:< −1 and take
148:
147:
144:
143:
140:
139:
445:
414:
413:
406:
356:Intrinsic metric
349:Intrinsic metric
196:, there is some
120:
119:
116:
113:
110:
89:
84:
83:
73:
66:
65:
55:
48:
31:
25:
24:
16:
453:
452:
448:
447:
446:
444:
443:
442:
423:
422:
411:
410:
404:
297:Oleg Alexandrov
249:Oleg Alexandrov
153:
117:
114:
111:
108:
107:
85:
78:
32:on Knowledge's
29:
12:
11:
5:
451:
449:
441:
440:
435:
425:
424:
421:
420:
400:
399:
398:
391:
388:
363:
362:
359:
352:
341:
340:
312:
310:
309:
308:
307:
293:
292:
291:
152:
149:
146:
145:
142:
141:
138:
137:
130:
124:
123:
121:
104:the discussion
91:
90:
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
450:
439:
436:
434:
431:
430:
428:
419:
415:
407:
401:
396:
392:
389:
386:
385:
383:
379:
378:
377:
376:
372:
368:
360:
357:
353:
350:
346:
345:
344:
339:
335:
331:
327:
326:
325:
324:
320:
316:
306:
302:
298:
294:
290:
286:
282:
277:
276:
275:
272:
268:
264:
260:
259:
258:
254:
250:
245:
244:
243:
242:
239:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
150:
135:
129:
126:
125:
122:
105:
101:
97:
96:
88:
82:
77:
75:
72:
68:
67:
63:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
405:siℓℓy rabbit
364:
342:
330:Sullivan.t.j
311:
271:Sullivan.t.j
262:
238:Sullivan.t.j
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
154:
93:
40:WikiProjects
109:Mathematics
100:mathematics
59:Mathematics
30:Start-class
427:Categories
351:is wrong)
236:−1) / 2.
172:\ with
180:. Then
263:caveat
36:scale.
367:Tosha
315:Tosha
412:talk
371:talk
334:talk
319:talk
301:talk
285:talk
253:talk
208:and
188:and
232:= (
216:= (
200:in
192:in
128:???
429::
416:)
402:--
373:)
336:)
321:)
303:)
287:)
255:)
220:+
168:=
160:,
408:(
369:(
358:)
332:(
317:(
299:(
283:(
251:(
234:x
230:z
226:x
222:y
218:x
214:z
210:y
206:x
202:X
198:z
194:X
190:y
186:x
182:X
178:R
174:d
170:R
166:X
162:d
158:X
156:(
136:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
