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Ian Calvert
Ian Calvert
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A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.

For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.

For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets , bounds as a function of m for compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed 3-convex sets in 1977. I know of no associated higher dimension work.

A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.

For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.

For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets , bounds as a function of m for compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed sets in 1977. I know of no associated higher dimension work.

A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.

For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.

For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets , bounds as a function of m for compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed 3-convex sets in 1977. I know of no associated higher dimension work.

Source Link
Ian Calvert
Ian Calvert
  • 313
  • 1
  • 12

A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.

For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.

For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets , bounds as a function of m for compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed sets in 1977. I know of no associated higher dimension work.

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