# Regular convex hull

In the field of n \ -C_ n _ * for a given givenregular convex hull(holmorphically convex hull) is defined as follows.

G ⊂ C n {\ displaystyle G \ subset {\ mathbb {C}} ^ {n}}! [G \ subset {\ \ mathbb {C}} ^ n] (https://wikimedia.org/api/rest_v1/media/math/render/svg/a669d69f08cb6302f1ba79303e9c98c14cb13083) (Ie) or more generally n {\ displaystyle n} ! [n] (https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) - dimension. O (G) {\ displaystyle {\ mathcal {O}} (G)}! [{\ \ Displaystyle {\ mathcal {O}} \ (G )}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/901a5bf6807d9f1966cc1562efc9afbe92801cc6) , G {\ displaystyle G} ! [G] (https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) It is a set of the above. A compact set K ⊂ G {\ displaystyle K \ subset G}! [{\ \ Displaystyle K \ subset G}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/80f658720077576ddab27442f5d32b8bcf0a7814) Regular convex hullis defined as follows.

For all f ∈ O (G)}. {\ Displaystyle {\ hat {K}} _ {G (k)} = {z ∈ G | f (z) | ≤ sup w ∈ K | f (z) \ right | \ leq \ sup _ {w \ in K} \ left | f (w) \ right | {\ mbox { {} displaystyle {\ hat {K}} _ {G}: = \ {z \ in G {\ \ big |} \ left | f \ (z ) \ right | \ leq \ sup _ {w \ in K} \ left | f \ (w ) \ right | {\ \ mbox {for all}} f \ \ in {\ \ mathcal {O}} \ (G ) \}.}] (https://wikimedia.org/api/rest_v1/media/math/render/svg / 496 b 239 64 7 9 1 f 8 c 34 e 6 d 37 b 32 8 f 88 48 c 81 f 424 6)

By setting _f _ in this definition, we obtain a more special conceptpolynomial convex hull(polynomial convex hull).

G {\ displaystyle G} ! [G] (https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) All compact in K ⊂ G {\ displaystyle K \ subset G}! [{\ \ Displaystyle K \ subset G}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/80f658720077576ddab27442f5d32b8bcf0a7814) K ^ G {\ displaystyle {\ hat {K}} _ {G}}! {{\ \ Displaystyle {\ hat {K}} _ {G}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/b431aa639633cda251f8997d0502e7fdfbe1f1aa) Also G {\ displaystyle G} ! [G] (https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) If it is compact within such area G {\ displaystyle G} ! [G] (https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) It is said to beregular hollow(holomorphically convex). This is often abbreviated holomorph-convex .

n = 1 {\ displaystyle n = 1} ! [n = 1] (https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425) K ^ G {\ displaystyle {\ hat {K}} _ {G}}! [{\ \ Displaystyle {\ hat {K}} _ {G}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/b431aa639633cda251f8997d0502e7fdfbe1f1aa) Is G ∖ K ⊂ G {\ displaystyle G \ setminus K \ subset G}! [{\ \ Displaystyle G \ setminus K \ subset G}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4e932b97398cbb80f91ec6b56cd926d47b7e59) Relative compact component and K {\ displaystyle K} ! [K] (https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0) Because it is a merger with arbitrary area G {\ displaystyle G} ! [G] (https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b) Is a regular convex. Also note that at this time, it is equivalent to that the area is regular convex (Carnot-Turren's theorem). These concepts are It is even more important when n> 1.

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References

. An Introduction to Complex Analysis in Several Variables , North-Holland Publishing Company, New York, New York, 1973. Steven G. Krantz. Function Theory of Several Complex Variables , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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Post Date : 2018-01-24 20:30