T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, 1987, pp. 135-149; MR 49 #9736. Home/Search Document Not in Database Summary Related Articles Check |
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Data Filtering and Distribution Modeling Algorithms for Machine .. - Yoav Freund (1993) (7 citations) (Correct)
....can say that the set of bodies S t , parameterized by t 2 R is a (one parameter) concave family of bodies. 9 The Brunn Minkowski theorem states that, for bodies in R n , the n th root of the volume of the bodies of a linear or concave family is a concave function of the family of parameters ([Bonnesen and Fenchel, 1987],Subsection 48) In our case, n = d Gamma 1 and the family is a concave family of a single parameter. We thus get the statement of the lemma as a special case of the Brunn Minkowski theorem. We shall use G d (r) to denote G(F ) where F is the volume function of a d dimensional body whose radius ....
....(3. 18) we get G(F d ) R 0 Gamma1 (1 t)d 1 2 (1 Gamma (1 t)d 1 2 )H( 1 t)d 1 2 ) dt R 0 Gamma1 (1 t)d 1 2 (1 Gamma (1 t)d 1 2 ) dt = R 1=2 0 F 1=d (1 Gamma F )H(F )dF R 1=2 0 F 1=d (1 Gamma F )dF ; 3:23) 9 For the definition of a convex family of bodies see ([Bonnesen and Fenchel, 1987],Subsection 24) 69 which can be shown by direct calculation to decrease as d 1. Which gives the general lower bound of G(F d ) R 1=2 0 (1 Gamma F )H(F )dF R 1=2 0 (1 Gamma F )dF = 1 9 7 18 log 2 : 3:24) And this gives the statement of the theorem. What remains to be shown is the ....
T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, Idaho, USA, 1987.
Selective sampling using the Query by Committee algorithm - Yoav Freund, H.. (1997) (55 citations) (Correct)
....say that the set of bodies S t , parameterized by t 2 R is a (one parameter) concave family of bodies. 12 The Brunn Minkowski theorem states that, for bodies in R n , the n th root of the volume of the bodies of a linear or concave family is a concave function of the family of parameters ((Bonnesen Fenchel,1987),Subsection 48) In our case, n = d Gamma 1 and the family is a concave family of a single parameter. We thus get the statement of the lemma as a special case of the Brunn Minkowski theorem. Proof of Lemma 6: As the value of the functional G(F ) is always positive, there must exist an infimum to ....
....derivative is defined and calculated can be found in standard books on variational analysis, such as (Smith,1985) 10. This definition is a one sided version of the notion of closeness defined in Definition 6.2. 11.Ignoring log factors. 12.For the definition of a convex family of bodies see ((Bonnesen Fenchel,1987),Subsection 24) 13.We use A Gamma B to denote the line segment between the points A and B, and A B to denote the segment of a curve that connects A and B. We also use the shorthand A Gamma B C Gamma D to denote a the concatenation of a line segment, a curve segment, and another line segment. ....
T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, Idaho, USA, 1987.
Reuleaux Triangle Constants - Finch (2003) (Correct)
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T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, 1987, pp. 135-149; MR 49 #9736.
Selective sampling using the Query by Committee algorithm - Freund, Seung, Shamir.. (1997) (55 citations) (Correct)
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T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, Idaho, USA, 1987.
How Many Zeros of a Random Polynomial are Real? - Edelman, Kostlan (1995) (1 citation) (Correct)
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T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, Idaho, 1987.
Selective sampling using the Query by Committee algorithm - Yoav Freund, H.. (1995) (55 citations) (Correct)
No context found.
T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, Idaho, USA, 1987.
How Many Zeros of a Random Polynomial are Real? - Edelman, Kostlan (1994) (1 citation) (Correct)
No context found.
T. Bonnesen and W. Fenchel, Theory of Convex Bodies. BCS Associates, Moscow, Idaho, 1987.
Selective sampling using the Query by Committee algorithm - Yoav Freund, H.. (1995) (55 citations) (Correct)
No context found.
T. Bonnesen and W. Fenchel. Theory of Convex Bodies. BCS Associates, Moscow, Idaho, USA, 1987.
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