Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

### The abstract of the Mizar article:

### Convex Hull, Set of Convex Combinations and Convex Cone

**by****Noboru Endou, and****Yasunari Shidama**- Received June 16, 2003
- MML identifier: CONVEX3

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, FUNCT_1, ARYTM_1, RELAT_1, FINSEQ_1, BOOLE, JORDAN1, ARYTM_3, CONVEX1, FINSEQ_4, SEQ_1, CARD_1, RLVECT_2, FUNCT_2, RFINSEQ, FINSET_1, CONVEX2, CONVEX3; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, FINSEQ_1, RELAT_1, FUNCT_1, FINSET_1, CARD_1, NAT_1, STRUCT_0, FUNCT_2, RLVECT_1, FINSEQ_4, RLVECT_2, RVSUM_1, CONVEX1, TOPREAL1, RFINSEQ, CONVEX2; constructors REAL_1, SEQ_1, RUSUB_5, FINSEQ_4, CONVEX1, NAT_1, MONOID_0, TOPREAL1, RFINSEQ, MEMBERED; clusters RELSET_1, ARYTM_3, STRUCT_0, RLVECT_1, SEQ_1, RUSUB_4, BINARITH, XREAL_0, MEMBERED, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Equality of Convex Hull and Set of Convex Combinations definition let V be RealLinearSpace; func ConvexComb(V) -> set means :: CONVEX3:def 1 for L being set holds L in it iff L is Convex_Combination of V; end; definition let V be RealLinearSpace, M be non empty Subset of V; func ConvexComb(M) -> set means :: CONVEX3:def 2 for L being set holds L in it iff L is Convex_Combination of M; end; theorem :: CONVEX3:1 for V being RealLinearSpace, v being VECTOR of V holds ex L being Convex_Combination of V st Sum(L) = v & (for A being non empty Subset of V st v in A holds L is Convex_Combination of A); theorem :: CONVEX3:2 for V being RealLinearSpace, v1,v2 being VECTOR of V st v1 <> v2 holds ex L being Convex_Combination of V st for A being non empty Subset of V st {v1,v2} c= A holds L is Convex_Combination of A; theorem :: CONVEX3:3 for V being RealLinearSpace, v1,v2,v3 being VECTOR of V st v1 <> v2 & v1 <> v3 & v2 <> v3 holds ex L being Convex_Combination of V st for A being non empty Subset of V st {v1,v2,v3} c= A holds L is Convex_Combination of A; theorem :: CONVEX3:4 for V being RealLinearSpace, M being non empty Subset of V holds M is convex iff {Sum(L) where L is Convex_Combination of M : L in ConvexComb(V)} c= M; theorem :: CONVEX3:5 for V being RealLinearSpace, M being non empty Subset of V holds conv(M) = {Sum(L) where L is Convex_Combination of M : L in ConvexComb(V)}; begin :: Cone and Convex Cone definition let V be non empty RLSStruct, M be Subset of V; attr M is cone means :: CONVEX3:def 3 for r being Real, v being VECTOR of V st r > 0 & v in M holds r*v in M; end; theorem :: CONVEX3:6 for V being non empty RLSStruct, M being Subset of V st M = {} holds M is cone; definition let V be non empty RLSStruct; cluster cone Subset of V; end; definition let V be non empty RLSStruct; cluster empty cone Subset of V; end; definition let V be RealLinearSpace; cluster non empty cone Subset of V; end; theorem :: CONVEX3:7 for V being non empty RLSStruct, M being cone Subset of V st V is RealLinearSpace-like holds M is convex iff for u,v being VECTOR of V st u in M & v in M holds u + v in M; theorem :: CONVEX3:8 for V being RealLinearSpace, M being Subset of V holds M is convex & M is cone iff (for L being Linear_Combination of M st Carrier L <> {} & for v being VECTOR of V st v in Carrier L holds L.v > 0 holds Sum(L) in M); theorem :: CONVEX3:9 for V being non empty RLSStruct, M,N being Subset of V st M is cone & N is cone holds M /\ N is cone;

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