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Each convex set containing X must ( by the assumption that it is convex ) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the sets given by these two definitions must be equal.

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## Some Related Sentences

Each and convex

__Each__point

**in**

**that**

**is**not

**in**

**the**

__convex__hull

**of**

**the**other

**points**

**(**

**that**

**is**

**,**such

**that**

**)**

**is**called

**a**vertex

**of**

**.**

__Each__

__convex__regular polychoron

**is**bounded

**by**

**a**

**set**

**of**3-dimensional cells which are

**all**Platonic solids

**of**

**the**same type

**and**size

**.**

Each and set

__Each__time Letch `` went up ''

**in**his `` lines ''

**,**I was

**the**one to

**be**patient

**,**helpful

**and**apologetic while he indulged

**in**outbursts

**of**temperament

**,**profanity

**and**abuse

**,**blaming others

**,**going into `` sulks ''

**and**

**,**on more occasions than I care to count

**,**storming off

**the**``

__set__'' for

**the**rest

**of**

**the**day

**.**

__Each__switch occupies one digit position

**in**

**a**word

**,**can

**be**

__set__on or off

**,**

**and**

**is**considered as logically equivalent to an electronic switch

**.**

__Each__element has

**a**specific

__set__

**of**chemical properties as

**a**consequence

**of**

**the**number

**of**electrons present

**in**

**the**neutral atom

**,**which

**is**Z

**(**

**the**atomic number ).

__Each__orbital

**is**defined

**by**

**a**different

__set__

**of**quantum numbers

**(**n

**,**l

**,**

**and**m ),

**and**

**contains**

**a**maximum

**of**

**two**electrons each with their own spin quantum number

**.**

__Each__cell site will use

**a**subset

**of**

**these**channels

**,**

**and**

**must**use

**a**different

__set__than neighboring cells to avoid interference

**.**

__Each__series was

__set__

**in**

**a**different historical period with

**the**

**two**protagonists accompanied

**by**different characters

**,**though several reappear

**in**one series or another

**,**for example Melchett

**and**Lord Flashheart

**.**

__Each__series was

__set__

**in**

**a**different period

**of**British history

**,**beginning

**in**1485

**and**ending

**in**1917

**,**

**and**comprised six half-hour episodes

**.**

__Each__sub-group

**of**

**two**interacting couples

**is**known to choreographers as

**a**minor

__set__

**and**to dancers as

**a**foursome or hands four

**.**

__Each__tile pattern

**in**

**the**Chinese domino

__set__

**is**made up

**of**

**the**outcome

**of**

**a**throw

**of**

**two**six-sided dice

**.**

__Each__province

**,**

**and**even some cities

**,**

__set__up its own autonomous juntas

**,**which declared themselves sovereign from each other

**.**

__Each__

**of**

**these**strings p < sub > i </ sub > determines

**a**subset S < sub > i </ sub >

**of**Cantor space ;

**the**

__set__S < sub > i </ sub >

**contains**

**all**sequences

**in**cantor space

**that**begin with p < sub > i </ sub >.

__Each__episode opened

**(**

**and**closed

**)**with

**a**short spoken introduction

**(**delivered

**by**Morris

**)**describing

**,**

**in**surreal

**,**broken language

**,**various bizarre feelings

**and**situations

**,**

__set__to ambient music interspersed with short clips

**of**other songs

**.**

__Each__sender has

**a**different

**,**unique vector v chosen from

**that**

__set__

**,**but

**the**construction method

**of**

**the**transmitted vector

**is**identical

**.**

__Each__progressively larger

__set__increases

**the**maximum number

**of**pips on an end

**by**three

**,**

**so**

**the**common extended

**sets**are double-nine

**,**double-twelve

**,**double-fifteen

**and**double-eighteen

**.**

__Each__

**of**

**the**columns from 3 to N

**is**

__set__to

**a**value derived from

**the**first

**and**higher derivatives

**of**

**the**polynomial

**.**

__Each__station transmitted

**a**continuous wave signal

**that**

**,**

**by**comparing

**the**phase difference

**of**

**the**signals from

**the**Master

**and**one

**of**

**the**Slaves

**,**resulted

**in**

**a**

__set__

**of**hyperbolic lines

**of**position called

**a**pattern

**.**

A partition

**of****X****is****a**__set__P**of**nonempty subsets**of****X****,**such**that**every element**of****X****is**an element**of****a**single element**of**P**.**__Each__element**of**P**is****a**cell**of****the**partition**.**

Each and containing

__Each__node maintains

**a**data store

__containing__documents associated with keys

**,**

**and**

**a**routing table associating nodes with records

**of**their performance

**in**retrieving different keys

**.**

__Each__country should have an accessible Aeronautical Information Publication

**(**AIP ), based on standards defined

**by**

**the**ICAO

**,**

__containing__information essential to air navigation

**.**

__Each__station displays

**the**Underground roundel

**,**often

__containing__

**the**station's name

**in**

**the**central bar

**,**at entrances

**and**repeatedly along

**the**platform

**,**

**so**

**that**

**the**name can easily

**be**seen

**by**passengers on arriving trains

**.**

__Each__society would maintain

**a**' master memex '

__containing__

**all**papers

**,**references

**,**tables " intimately interconnected

**by**trails

**,**

**so**

**that**one may follow

**a**detailed matter from paper to paper

**,**going back through

**the**classics

**,**recording criticism

**in**

**the**margins

**.**

__Each__player remaining

**in**

**the**game takes

**two**chips or coins below

**the**table

**,**then brings up

**a**closed hand

__containing__zero

**,**one

**,**or

**two**

**of**

**the**chips

**.**

__Each__Unitarian congregation

**is**at liberty to devise its own form

**of**worship

**,**though commonly

**,**Unitarian services lack liturgy

**and**ritual

**,**while

__containing__readings from many sources

**,**which may include sermons

**,**prayers

**,**hymns

**and**songs

**.**

__Each__mature Volvox colony

**is**composed

**of**numerous flagellate cells similar to Chlamydomonas

**,**up to 50

**,**000

**in**total

**,**

**and**embedded

**in**

**the**surface

**of**

**a**hollow sphere or coenobium

__containing__an extracellular matrix made

**of**

**a**gelatinous glycoprotein

**.**

__Each__whitish seed has

**a**small

**,**fleshy tail

**(**elaiosome

**)**

__containing__substances attractive to ants which distribute

**the**seeds

**.**

__Each__cnidocyte cell

**contains**an organelle called

**a**cnidae or cnidocyst

**(**e

**.**g

**.**nematocyst

**,**ptychocyst or spirocyst ), which comprises

**a**bulb-shape capsule

__containing__

**a**coiled hollow tubule structure attached to

**it**

**.**

**(**

**a**

**)**Certification

**of**Periodic Financial Reports .—

__Each__periodic report

__containing__financial statements filed

**by**an issuer with

**the**Securities Exchange Commission pursuant to section 13

**(**

**a**

**)**or 15

**(**d

**)**

**of**

**the**Securities Exchange Act

**of**1934

**(**15 U

**.**S

**.**C

**.**

__Each__lararium features

**a**panel fresco

__containing__

**the**same theme:

**two**peripheral figures

**(**Lares

**)**attend on

**a**central figure

**(**family genius

**)**or

**two**figures

**(**genius

**and**Juno

**)**who may or may not

**be**at an altar

**.**

__Each__page

__containing__Google advertisements adds

**,**reads

**,**

**and**modifies " cookies " on each visitor's computer

**.**

__Each__page

__containing__Google advertisements adds

**,**reads

**,**

**and**modifies " cookies " on each visitor's computer

**.**

Artists included Alexander Kipnis

**(**III**,**IV**,**V ); Herbert Janssen**(**II**,**V ); Gerhard Hüsch**(**II**,**III**,**IV**,**V ); John McCormack**(**accompanied**by**Edwin Schneider**)****(**II ); Alexandre Trianti**(**II**,**III ); Ria Ginster**(**IV**,**V ); Friedrich Schorr**(**II ); Elisabeth Rethberg**(**IV**,**V ); Tiana Lemnitz__Each__volume was accompanied**by****a**booklet__containing__**a**short essay**by**Ernest Newman**(**I: Words**and**Music**in**Hugo Wolf**,**II: Wolf's Goethe Songs**,**III: A Note**of**Wolf as Craftsman**,**IV: The Italienisches Liederbuch**)**together with German texts**,**English translations**(****by**Winifred Radford**)****and**notes on each song**(****by**Newman ).__Each__mound covered

**a**single grave

**,**

__containing__

**a**coffin made from

**a**single split

**and**hollowed oak tree

**.**

__Each__album

**contained**

**two**seven-inch records

**,**plus

**a**fully illustrated booklet

__containing__

**the**text

**of**

**the**recording

**,**

**so**

**that**children could follow along

**by**reading

**.**

__Each__plexus consists

**of**

**a**singular artery

__containing__warm blood from

**the**heart surrounded

**by**

**a**bundle

**of**veins

__containing__cool blood from

**the**body surface

**.**

0.223 seconds.