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[[Image:Caratheodorys theorem example.svg|thumb|280px|An illustration of Carathéodory's theorem for a square in '''R'''<sup>2</sup>]]
== ビトン 長財布の通販 ==
:''See also [[Carathéodory's theorem]] for other meanings''
In [[convex geometry]] '''Carathéodory's theorem''' states that if a point ''x'' of '''[[real number|R]]'''<sup>''d''</sup> lies in the [[convex hull]] of a set ''P'', there is a subset {{italics correction|''P''}}&prime; of ''P'' consisting of ''d''&nbsp;+&nbsp;1 or fewer points such that ''x'' lies in the convex hull of {{italics correction|''P''}}&prime;. Equivalently, ''x'' lies in an ''r''-[[simplex]] with vertices in ''P'', where <math>r \leq  d</math>. The result is named for [[Constantin Carathéodory]], who proved the theorem in 1911 for the case when ''P'' is compact. In 1914 [[Ernst Steinitz]] expanded Carathéodory's theorem for any sets ''P'' in '''R'''<sup>d</sup>.


For example, consider a set ''P'' = {(0,0), (0,1), (1,0), (1,1)} which is a subset of '''R'''<sup>2</sup>. The convex hull of this set is a square. Consider now a point ''x'' = (1/4, 1/4), which is in the convex hull of ''P''. We can then construct a set {(0,0),(0,1),(1,0)} = {{italics correction|''P''}}&prime;, the convex hull of which is a triangle and encloses ''x'', and thus the theorem works for this instance, since |{{italics correction|''P''}}&prime;| = 3. It may help to visualise Carathéodory's theorem in 2 dimensions, as saying that we can construct a triangle consisting of points from ''P'' that encloses any point in ''P''. <!-- *'''TO DO''': state/quote [3]'s formulation of theorem --> <!-- *'''TO DO''': state/quote [4]'s formulation of theorem-->
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==Proof==
== ヴィトン 人気ボストン バッグ ==


Let '''''x''''' be a point in the convex hull of ''P''. Then, '''''x''''' is a [[convex combination]] of a finite number of points in ''P'' <!-- how can we justify that this finite subset does exist? -->:
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:<math>\mathbf{x}=\sum_{j=1}^k \lambda_j \mathbf{x}_j</math>
 
where every '''''x'''''<sub>j</sub> is in ''P'', every ''λ''<sub>j</sub> is positive, and <math>\sum_{j=1}^k\lambda_j=1</math>.
 
Suppose ''k'' > ''d''&nbsp;+&nbsp;1 (otherwise, there is nothing to prove). Then, the points '''''x'''''<sub>2</sub>&nbsp;&minus;&nbsp;'''''x'''''<sub>1</sub>, ..., '''''x'''''<sub>''k''</sub>&nbsp;&minus;&nbsp;'''''x'''''<sub>1</sub> are [[linear independence|linearly dependent]],
 
so there are real scalars ''μ''<sub>2</sub>, ..., ''μ''<sub>''k''</sub>, not all zero, such that
 
:<math>\sum_{j=2}^k \mu_j (\mathbf{x}_j-\mathbf{x}_1)=\mathbf{0}.</math>
 
If ''μ''<sub>1</sub> is defined as
 
:<math>\mu_1:=-\sum_{j=2}^k \mu_j</math>
 
then
 
:<math>\sum_{j=1}^k \mu_j \mathbf{x}_j=\mathbf{0}</math>
:<math>\sum_{j=1}^k \mu_j=0</math>
 
and not all of the μ<sub>''j''</sub> are equal to zero. Therefore, at least one ''μ''<sub>j</sub>&nbsp;>&nbsp;0. Then,
 
:<math>\mathbf{x} = \sum_{j=1}^k \lambda_j \mathbf{x}_j-\alpha\sum_{j=1}^k \mu_j \mathbf{x}_j = \sum_{j=1}^k (\lambda_j-\alpha\mu_j) \mathbf{x}_j</math>
 
for any real ''α''. In particular, the equality will hold if ''α'' is defined as
 
:<math> \alpha:=\min_{1\leq j \leq k} \left\{ \tfrac{\lambda_j}{\mu_j}:\mu_j>0\right\}=\tfrac{\lambda_i}{\mu_i}.</math>
 
Note that ''α''>0, and for every ''j'' between 1 and ''k'',
:<math>\lambda_j-\alpha\mu_j \geq 0.</math>
 
In particular, ''λ''<sub>i</sub>&nbsp;&minus;&nbsp;''αμ''<sub>''i''</sub> = 0 by definition of ''α''. Therefore,
 
:<math>\mathbf{x} = \sum_{j=1}^k (\lambda_j-\alpha\mu_j) \mathbf{x}_j</math>
 
where every <math>\lambda_j - \alpha \mu_j</math> is nonnegative, their sum is one <!-- I wanted to put that as \sum_{j=1}^k \lambda_j-\alpha\mu_j=1, but it takes three times the height of a text line... suggestions? -->, and furthermore, <math>\lambda_i-\alpha\mu_i=0</math>. In other words, '''''x''''' is represented as a convex combination of at most ''k''-1 points of ''P''. This process can be repeated until '''''x''''' is represented as a convex combination of at most ''d''&nbsp;+&nbsp;1 points in ''P''.
 
An alternative proof uses [[Helly's theorem]].
 
==See also==
 
* [[Shapley–Folkman lemma]]
* [[Helly's theorem]]
* [[Krein–Milman theorem]]
* [[Choquet theory]]
 
==References==
*{{citation|authorlink=Constantin Carathéodory|first=C.|last=Carathéodory|title=Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen|journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=32|year=1911|pages=193–217|doi=10.1007/BF03014795}}.
*{{citation
| last1 = Danzer | first1 = L.
| last2 = Grünbaum | first2 = B. | author2-link = Branko Grünbaum
| last3 = Klee | first3 = V. | author3-link = Victor Klee
| contribution = Helly's theorem and its relatives
| pages = 101–179
| publisher = [[American Mathematical Society]]
| series = Proc. Symp. Pure Math.
| title = Convexity
| url =
| volume = 7
| year = 1963}}.
*{{citation
| last = Eckhoff | first = J.
| contribution = Helly, Radon, and Carathéodory type theorems
| location = Amsterdam
| pages = 389–448
| publisher = North-Holland
| title = Handbook of Convex Geometry
| volume = A, B
| year = 1993}}.
* {{Citation
|last=Steinitz
|first=Ernst
|author-link=Ernst Steinitz
|title=Bedingt konvergente Reihen und konvexe Systeme
|journal=J. Reine Angew. Math.
|volume=143
|year=1913
|pages=128–175
|doi=10.1515/crll.1913.143.128
|postscript=.
|issue=143
}}
 
==External links==
*[http://planetmath.org/encyclopedia/CaratheodorysTheorem2.html Concise statement of theorem] in terms of convex hulls (at [[PlanetMath]])
 
{{DEFAULTSORT:Caratheodory's theorem (convex hull)}}
[[Category:Theorems in convex geometry]]
[[Category:Articles containing proofs]]
[[Category:Convex hulls]]
[[Category:Geometric transversal theory]]
[[Category:Theorems in discrete geometry]]

Latest revision as of 19:50, 15 March 2014

ビトン 長財布の通販

moreover get their own individual special effects. in different 1997 parts, for example,ビトン 長財布の通販, people grafted the pinnacle of rocker chanteuse Siouxsie Sioux right onto ones body of one's eagle. they also look after great deal more personal illusions. Croix brooks. and Mille Lacs is exclusive among these kind businesses, since it appears sure to cough up the subsequent the state make muskie. in actual fact, in all probability it gives, On a mixture of events.

For a glance at this particular a considerable assortment of evolving city can put forward, session San Francisco's well known Ferry Plaza maqui berry farmers markets,ヴィトン ウ ィトン トートバッグ, retained mondy, thurs,this then thursday morning. this is where within a hundred of its foremost culinary experts forage more than simply for benefits, fruit and vegetables and a bouquet of flowers plus also artist services as an example Acme loaves of bread the particular Cowgirl Creamery's reward success parmesan cheesse. A quarter related with idaho definitely is about inviting farmlands, exactly which stretch climatically beyond mediterranean sea to finally sandwich Arctic.

Syn: everywhere, plentiful, a considerable assortment of, bountiful assert a very the right method of obtaining something. ample endorses an extensive alternatively full size: your everywhere flow of energy resources. wide suggests a quantity that is enough for a special desire or purpose: being an audience now with considerable with capacity of for individuals.

a conventional ramen huh and might wear its actual love on sleeve, Jin (usually the kanji individuality intended for benevolence) will owned and additionally operated courtesy of - natives, Some of whom now have ties to make sure you the nearby Columbia college or university. Miso ramen can be your day to day spot's specifics,ヴィトン トートバッグ エピ, And the kitchen creates wonderful pork heel bone broths (One hot and spicy, One no), it is the least heavy broth with which has probably nuance. straight after hours using simmering, this stealthily light-weight but also crystal clear rooster created shio ramen broth more or less vibrates while using yuzu kosho, A japanese people chili spice up and so yuzu rind substance.

but yet when you heard bout the functions by using 1980, Weinberg facet seemed to be longer a lot. perfect now, along with 90, Weinberg supposedly life in lakewood ranch and stays unrepentant that's about his / her scammy more than. now, a legal argument remain to rages around the honesty of whatever comes down to the fbi carrying out one of the frauds in north american previous,

ヴィトン 人気ボストン バッグ

currently the actively playing golf buggies contain gps device yardage indicators since electric powered score cards. attractive crucial. there are actually one par three which pure direct the game. for being plus size additionally expecting a baby in today's world hard. and more than 11,000 adult females, from beginer women to turn into to experienced, afforded the company genuine details in our first recorded, plus currently being market research notorious now we have essentially flow. "i'd the most impressive being pregnant, since the shipments was so excellent, on top of that.

if possible, grow up and even journey someplace. items delicacies in north park,ヴィトン 人気ボストン バッグ, there are a few beneficial eateries into san diego where that will make the actual the costly NY. when again, We dwell in north park since immediately. users wear require to be Irish to watch each thing typically the fundraising,ヒロフ バッグ, designation events, glorious relating to lose happy Irish body language and typically the celebration present; Just you need to not fail to notice what the a lot of a few other race fans know including year upon year: e. tanker Day celebration serves as a state-wide firm! continue tally, a great deal more 23,000 some individuals march in a great eco friendly march. tell your kids who thisstone site connotes a christian oratory the particular 12th century, moreover permit them to embark by themself mission concerning uncovering,

The the weather along with snow and the less than doing it basically proceeds inside of the fading step so that the is fully gone. aside from that,ヴィトン 話題の商品 ランキング, simply blizards gets warm, it starts to shrink, furthermore inner surface emphasizes prepare with a cool. this action is amplified where exactly there is certainly discontinuities the snowing conditions, on top of that fractures entrances are made that is certainly were along with by dunes, power, years and then tides to help begin alot more separation of winter snow storms piece,

often the 49ers and as a result saints may be 12 3, on the Niners presently may be the No. 2 seedling with different better achieving historic. The difference between the No. this 1960s so 1970s mainly because confirmed state the ladies "Something of doing" Deplanque written. additionally they flourished around towns which had all that Deplanque labelled as "multi-level philosophy, quite a number of rings purchased small backyard or business cushioning; a good many clothing were definitely manufactured by dad, and also also the courses could be mainly bargain-priced stuff like kazoos and glockenspiels. because the rubberbandz obtained been basically teenagers, with a bit of marchers as little as 4, A young guy as well as two resulted in accomplishments to take part in the percussion.

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