:: Convex Sets and Convex Combinations
:: by Noboru Endou , Takashi Mitsuishi and Yasunari Shidama
::
:: Received November 5, 2002
:: Copyright (c) 2002 Association of Mizar Users

begin

definition

let V be non empty RLSStruct ;
let M be Subset of V;
let r be Real;
func r * M -> Subset of V equals :: CONVEX1:def 1
{ (r * v) where v is Element of V : v in M } ;
coherence
{ (r * v) where v is Element of V : v in M } is Subset of V

proof end;

end;

:: deftheorem defines * CONVEX1:def 1 :
for V being non empty RLSStruct
for M being Subset of V
for r being Real holds r * M = { (r * v) where v is Element of V : v in M } ;

definition

let V be non empty RLSStruct ;
let M be Subset of V;
attr M is convex means :Def2: :: CONVEX1:def 2
for u, v being VECTOR of V
for r being Real st 0 < r & r < 1 & u in M & v in M holds
(r * u) + ((1 - r) * v) in M;

end;

:: deftheorem Def2 defines convex CONVEX1:def 2 :
for V being non empty RLSStruct
for M being Subset of V holds
( M is convex iff for u, v being VECTOR of V
for r being Real st 0 < r & r < 1 & u in M & v in M holds
(r * u) + ((1 - r) * v) in M );

theorem :: CONVEX1:1

for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V
for r being Real st M is convex holds
r * M is convex

proof end;

theorem :: CONVEX1:2

for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M, N being Subset of V st M is convex & N is convex holds
M + N is convex

proof end;

theorem :: CONVEX1:3

for V being RealLinearSpace
for M, N being Subset of V st M is convex & N is convex holds
M - N is convex

proof end;

theorem Th4: :: CONVEX1:4

for V being non empty RLSStruct
for M being Subset of V holds
( M is convex iff for r being Real st 0 < r & r < 1 holds
(r * M) + ((1 - r) * M) c= M )

proof end;

theorem :: CONVEX1:5

for V being non empty Abelian RLSStruct
for M being Subset of V st M is convex holds
for r being Real st 0 < r & r < 1 holds
((1 - r) * M) + (r * M) c= M

proof end;

theorem :: CONVEX1:6

for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M, N being Subset of V st M is convex & N is convex holds
for r being Real holds (r * M) + ((1 - r) * N) is convex

proof end;

Lm1: for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V holds 1 * M = M

proof end;

Lm2: for V being RealLinearSpace
for M being non empty Subset of V holds 0 * M = {(0. V)}

proof end;

Lm3: for V being non empty add-associative addLoopStr
for M1, M2, M3 being Subset of V holds (M1 + M2) + M3 = M1 + (M2 + M3)

proof end;

Lm4: for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V
for r1, r2 being Real holds r1 * (r2 * M) = (r1 * r2) * M

proof end;

Lm5: for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M1, M2 being Subset of V
for r being Real holds r * (M1 + M2) = (r * M1) + (r * M2)

proof end;

theorem :: CONVEX1:7

for V being RealLinearSpace
for M being Subset of V
for v being VECTOR of V holds
( M is convex iff v + M is convex )

proof end;

theorem :: CONVEX1:8

for V being RealLinearSpace holds Up ((0). V) is convex

proof end;

theorem Th9: :: CONVEX1:9

for V being RealLinearSpace holds Up ((Omega). V) is convex

proof end;

theorem Th10: :: CONVEX1:10

for V being non empty RLSStruct
for M being Subset of V st M = {} holds
M is convex

proof end;

theorem Th11: :: CONVEX1:11

for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M1, M2 being Subset of V
for r1, r2 being Real st M1 is convex & M2 is convex holds
(r1 * M1) + (r2 * M2) is convex

proof end;

theorem Th12: :: CONVEX1:12

for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V
for r1, r2 being Real holds (r1 + r2) * M c= (r1 * M) + (r2 * M)

proof end;

Lm6: for V being non empty RLSStruct
for M, N being Subset of V
for r being Real st M c= N holds
r * M c= r * N

proof end;

Lm7: for V being non empty RLSStruct
for M being empty Subset of V
for r being Real holds r * M = {}

proof end;

Lm8: for V being non empty addLoopStr
for M being empty Subset of V
for N being Subset of V holds M + N = {}

proof end;

Lm9: for V being non empty right_zeroed addLoopStr
for M being Subset of V holds M + {(0. V)} = M

proof end;

Lm10: for V being RealLinearSpace
for M being Subset of V
for r1, r2 being Real st r1 >= 0 & r2 >= 0 & M is convex holds
(r1 * M) + (r2 * M) c= (r1 + r2) * M

proof end;

theorem :: CONVEX1:13

for V being RealLinearSpace
for M being Subset of V
for r1, r2 being Real st r1 >= 0 & r2 >= 0 & M is convex holds
(r1 * M) + (r2 * M) = (r1 + r2) * M

proof end;

theorem :: CONVEX1:14

for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M1, M2, M3 being Subset of V
for r1, r2, r3 being Real st M1 is convex & M2 is convex & M3 is convex holds
((r1 * M1) + (r2 * M2)) + (r3 * M3) is convex

proof end;

theorem Th15: :: CONVEX1:15

for V being non empty RLSStruct
for F being Subset-Family of V st ( for M being Subset of V st M in F holds
M is convex ) holds
meet F is convex

proof end;

theorem Th16: :: CONVEX1:16

for V being non empty RLSStruct
for M being Subset of V st M is Affine holds
M is convex

proof end;

registration

let V be non empty RLSStruct ;
cluster non empty convex Element of K10( the carrier of V);
existence
ex b₁ being Subset of V st
( not b₁ is empty & b₁ is convex )

proof end;

end;

registration

let V be non empty RLSStruct ;
cluster empty convex Element of K10( the carrier of V);
existence
ex b₁ being Subset of V st
( b₁ is empty & b₁ is convex )

proof end;

end;

theorem :: CONVEX1:17

for V being non empty RealUnitarySpace-like UNITSTR
for M being Subset of V
for v being VECTOR of V
for r being Real st M = { u where u is VECTOR of V : u .|. v >= r } holds
M is convex

proof end;

theorem :: CONVEX1:18

for V being non empty RealUnitarySpace-like UNITSTR
for M being Subset of V
for v being VECTOR of V
for r being Real st M = { u where u is VECTOR of V : u .|. v > r } holds
M is convex

proof end;

theorem :: CONVEX1:19

for V being non empty RealUnitarySpace-like UNITSTR
for M being Subset of V
for v being VECTOR of V
for r being Real st M = { u where u is VECTOR of V : u .|. v <= r } holds
M is convex

proof end;

theorem :: CONVEX1:20

for V being non empty RealUnitarySpace-like UNITSTR
for M being Subset of V
for v being VECTOR of V
for r being Real st M = { u where u is VECTOR of V : u .|. v < r } holds
M is convex

proof end;

begin

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;
attr L is convex means :Def3: :: CONVEX1:def 3
ex F being FinSequence of the carrier of V st
( F is one-to-one & rng F = Carrier L & ex f being FinSequence of REAL st
( len f = len F & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (F . n) & f . n >= 0 ) ) ) );

end;

:: deftheorem Def3 defines convex CONVEX1:def 3 :
for V being RealLinearSpace
for L being Linear_Combination of V holds
( L is convex iff ex F being FinSequence of the carrier of V st
( F is one-to-one & rng F = Carrier L & ex f being FinSequence of REAL st
( len f = len F & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (F . n) & f . n >= 0 ) ) ) ) );

theorem Th21: :: CONVEX1:21

for V being RealLinearSpace
for L being Linear_Combination of V st L is convex holds
Carrier L <> {}

proof end;

theorem :: CONVEX1:22

for V being RealLinearSpace
for L being Linear_Combination of V
for v being VECTOR of V st L is convex & L . v <= 0 holds
not v in Carrier L

proof end;

theorem :: CONVEX1:23

for V being RealLinearSpace
for L being Linear_Combination of V st L is convex holds
L <> ZeroLC V

proof end;

Lm11: for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v} holds
( L . v = 1 & Sum L = (L . v) * v )

proof end;

Lm12: for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2} & v1 <> v2 holds
( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )

proof end;

Lm13: for p being FinSequence
for x, y, z being set st p is one-to-one & rng p = {x,y,z} & x <> y & y <> z & z <> x & not p = <*x,y,z*> & not p = <*x,z,y*> & not p = <*y,x,z*> & not p = <*y,z,x*> & not p = <*z,x,y*> holds
p = <*z,y,x*>

proof end;

Lm14: for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 holds
Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3)

proof end;

Lm15: for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2,v3} & v1 <> v2 & v2 <> v3 & v3 <> v1 holds
( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )

proof end;

theorem :: CONVEX1:24

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of {v} st L is convex holds
( L . v = 1 & Sum L = (L . v) * v )

proof end;

theorem :: CONVEX1:25

for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of {v1,v2} st v1 <> v2 & L is convex holds
( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )

proof end;

theorem :: CONVEX1:26

for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 & L is convex holds
( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )

proof end;

theorem :: CONVEX1:27

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v} holds
L . v = 1 by Lm11;

theorem :: CONVEX1:28

for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2} & v1 <> v2 holds
( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 ) by Lm12;

theorem :: CONVEX1:29

for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2,v3} & v1 <> v2 & v2 <> v3 & v3 <> v1 holds
( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) ) by Lm15;

begin

definition

let V be non empty RLSStruct ;
let M be Subset of V;
func Convex-Family M -> Subset-Family of V means :Def4: :: CONVEX1:def 4
for N being Subset of V holds
( N in it iff ( N is convex & M c= N ) );
existence
ex b₁ being Subset-Family of V st
for N being Subset of V holds
( N in b₁ iff ( N is convex & M c= N ) )

proof end;

uniqueness
for b₁, b₂ being Subset-Family of V st ( for N being Subset of V holds
( N in b₁ iff ( N is convex & M c= N ) ) ) & ( for N being Subset of V holds
( N in b₂ iff ( N is convex & M c= N ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Convex-Family CONVEX1:def 4 :
for V being non empty RLSStruct
for M being Subset of V
for b₃ being Subset-Family of V holds
( b₃ = Convex-Family M iff for N being Subset of V holds
( N in b₃ iff ( N is convex & M c= N ) ) );

definition

let V be non empty RLSStruct ;
let M be Subset of V;
func conv M -> convex Subset of V equals :: CONVEX1:def 5
meet (Convex-Family M);
coherence
meet (Convex-Family M) is convex Subset of V

proof end;

end;

:: deftheorem defines conv CONVEX1:def 5 :
for V being non empty RLSStruct
for M being Subset of V holds conv M = meet (Convex-Family M);

theorem :: CONVEX1:30

for V being non empty RLSStruct
for M being Subset of V
for N being convex Subset of V st M c= N holds
conv M c= N

proof end;

begin

theorem :: CONVEX1:31

for p being FinSequence
for x, y, z being set st p is one-to-one & rng p = {x,y,z} & x <> y & y <> z & z <> x & not p = <*x,y,z*> & not p = <*x,z,y*> & not p = <*y,x,z*> & not p = <*y,z,x*> & not p = <*z,x,y*> holds
p = <*z,y,x*> by Lm13;

theorem :: CONVEX1:32

for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V holds 1 * M = M by Lm1;

theorem :: CONVEX1:33

for V being non empty RLSStruct
for M being empty Subset of V
for r being Real holds r * M = {} by Lm7;

theorem :: CONVEX1:34

for V being RealLinearSpace
for M being non empty Subset of V holds 0 * M = {(0. V)} by Lm2;

theorem :: CONVEX1:35

for V being non empty right_zeroed addLoopStr
for M being Subset of V holds M + {(0. V)} = M by Lm9;

theorem :: CONVEX1:36

for V being non empty add-associative addLoopStr
for M1, M2, M3 being Subset of V holds (M1 + M2) + M3 = M1 + (M2 + M3) by Lm3;

theorem :: CONVEX1:37

for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M being Subset of V
for r1, r2 being Real holds r1 * (r2 * M) = (r1 * r2) * M by Lm4;

theorem :: CONVEX1:38

for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for M1, M2 being Subset of V
for r being Real holds r * (M1 + M2) = (r * M1) + (r * M2) by Lm5;

theorem :: CONVEX1:39

for V being non empty RLSStruct
for M, N being Subset of V
for r being Real st M c= N holds
r * M c= r * N by Lm6;

theorem :: CONVEX1:40

for V being non empty addLoopStr
for M being empty Subset of V
for N being Subset of V holds M + N = {} by Lm8;

begin

registration

let V be non empty RLSStruct ;
let M, N be convex Subset of V;
cluster M /\ N -> convex Subset of V;
coherence
for b₁ being Subset of V st b₁ = M /\ N holds
b₁ is convex

proof end;

end;

registration

let V be RealLinearSpace;
let M be Subset of V;
cluster Convex-Family M -> non empty ;
coherence
not Convex-Family M is empty

proof end;

end;

theorem :: CONVEX1:41

for V being RealLinearSpace
for M being Subset of V holds M c= conv M

proof end;