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"In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics. The two ends of a line segment determine the points in between: in vector terms the segment from v to w consists of the λv + (1 − λ)w with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C is a (finite) convex combination of points e of E. Here E may be a finite or an infinite set. In vector terms, by assigning non-negative weights w(e) to the e in E, almost all 0, we can represent any c in C as : c = \sum_{e\in E} w(e) e\ with : \sum_{e\in E} w(e) = 1.\ In any case the w(e) give a probability measure supported on a finite subset of E. For any affine function f on C, its value at the point c is :f (c) = \int f(e) d w(e). In the infinite dimensional setting, one would like to make a similar statement. Choquet's theorem states that for a compact convex subset C of a normed space V, given c in C there exists a probability measure w supported on the set E of extreme points of C such that, for any affine function f on C, :f (c) = \int f(e) d w(e). In practice V will be a Banach space. The original Krein–Milman theorem follows from Choquet's result. Another corollary is the Riesz representation theorem for states on the continuous functions on a metrizable compact Hausdorff space. More generally, for V a locally convex topological vector space, the Choquet–Bishop–de Leeuw theoremErrett Bishop; Karl de Leeuw. "The representations of linear functionals by measures on sets of extreme points". Annales de l'Institut Fourier, 9 (1959), pp. 305–331. gives the same formal statement. In addition to the existence of a probability measure supported on the extreme boundary that represents a given point c, one might also consider the uniqueness of such measures. It is easy to see that uniqueness does not hold even in the finite dimensional setting. One can take, for counterexamples, the convex set to be a cube or a ball in R3. Uniqueness does hold, however, when the convex set is a finite dimensional simplex. A finite dimensional simplex is a special case of a Choquet simplex. Any point in a Choquet simplex is represented by a unique probability measure on the extreme points. See also * Carathéodory's theorem * Shapley–Folkman lemma * Krein–Milman theorem * Helly's theorem * List of convexity topics Notes References * Category:Functional analysis Category:Integral representations Category:Convex hulls "
"Daniel "Daan" Wepener Bekker (9 February 1932 – 22 October 2009) was a South African boxer, who won the bronze medal in the Heavyweight division (+ 91 kg) at the 1956 Summer Olympics in Melbourne, Australia. Four years later in Rome he captured the silver medal in the same category. Amateur career Bekker was the South African Heavyweight Champion from 1955–1959 and 1961. =Amateur results= *1956 won the Heavyweight bronze medal at the Melbourne Olympics. **Defeated José Giorgetti (Argentina) KO-1 **Lost to Pete Rademacher (USA) KO-3 *1958 won the Commonwealth Games in Cardiff, Wales. **Defeated S. Renaud (Canada) TKO-1 **Defeated Gbadegesin Salawu (Nigeria) TKO-2 **Defeated David Thomas (England) PTS *1960 won the Heavyweight silver medal at the Rome Olympics. **Defeated Władysław Jędrzejewski (Poland) TKO-1 **Defeated Obrad Sretenovic (Yugoslavia) KO-1 **Defeated Günter Siegmund (United Team of Germany) PTS (4-1) **Lost to Franco De Piccoli (Italy) KO-2 Death Bekker died on 18 October 2009 after a long struggle with Parkinson's and Alzheimer's diseases.Ron Jackson, SA super amateur passes Fightnews.com. Retrieved 23 October 2009. References External links * Category:1932 births Category:2009 deaths Category:People from Dordrecht, Eastern Cape Category:Heavyweight boxers Category:Deaths from Parkinson's disease Category:Deaths from Alzheimer's disease Category:Olympic boxers of South Africa Category:Boxers at the 1956 Summer Olympics Category:Boxers at the 1960 Summer Olympics Category:Olympic silver medalists for South Africa Category:Olympic bronze medalists for South Africa Category:Sportspeople from Pretoria Category:Commonwealth Games gold medallists for South Africa Category:Boxers at the 1958 British Empire and Commonwealth Games Category:Olympic medalists in boxing Category:Medalists at the 1956 Summer Olympics Category:Medalists at the 1960 Summer Olympics Category:Afrikaner people Category:South African male boxers Category:Commonwealth Games medallists in boxing "
"Marco Traniello (born August 14, 1973, in Gaeta) is an Italian professional hairdresser and is sponsored professional poker player on Full Tilt Poker. Poker Traniello made his first appearance at the World Series of Poker (WSOP) in 2005, finishing in the money in seven different events (tied for the most money finishes in the 2005 WSOP with Michael Mizrachi and Tony Cousineau). Amongst these money finishes, Traniello made the final table of the $2,000 pot limit hold'em event. He played in the 2005 World Speed Poker Open in London and cashed in the $15,000 no limit hold'em main event of the World Poker Tour Five Diamond World Poker Classic. Traniello writes for Card Player Magazine and is a member of Marcel Luske's Circle of Outlaws. At the 2007 World Series of Poker, he made it to the final table in the $10,000 Pot Limit Omaha event. He finished 5th, winning $156,435, his biggest win to date. As of January 2015, his total live tournament winnings exceed $700,000. His 27 cashes at the WSOP account for $489,220 of those winnings.World Series of Poker Earnings , worldseriesofpoker.com Personal life Prior to working as a professional poker player, Traniello was a real estate agent and the captain of an oil ship. References External links *Hendon Mob tournament results *Official Website *Official Twitter *Official FaceBook page Category:Italian poker players Category:Living people Category:1973 births "