1. The Dual Brunn-Minkowski Theory and Some of Its Inequalities
Richard Gardner
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2. Isoperimetric Inequality
If K is a convex body and B is the unit ball in En, then
with equality if and only if K is a ball.
The (n-1)st intrinsic volume Vn-1(K) is half the surface area S(K) of K.
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3. Dual Isoperimetric Inequality
If K is a convex body containing the origin in En, then
with equality if and only if K is an origin-symmetric ball.
The (n-1)st dual volume is …
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4. Dual Volumes
ith dual volume of a star body K is
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5. Relations Between Inequalities
R.J.G., The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002),
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6. Aleksandrov-Fenchel Inequality
If K1, K2,…, Kn are compact convex sets in En, then
Dual Aleksandrov-Fenchel Inequality
If K1, K2,…, Kn are star bodies in En containing the origin in their interiors, then
with equality when if and only if K1, K2,…, Ki are dilatates of each other.
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7. Dual Mixed Volumes
The dual mixed volume of star bodies K1, K2,…, Kn is
Proof of dual Aleksandrov-Fenchel inequality follows directly from an extension of Hölder’s inequality.
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975),
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8. Minkowski and Radial Addition
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9. Brunn-Minkowski inequality
If K and L are convex bodies in En, then
(B-M)
and
(M1)
with equality iff K and L are homothetic.
Dual Brunn-Minkowski inequality
If K and L are star bodies in En, then
(d.B-M)
and
(d.M1)
with equality iff K and L are dilatates.
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10. Relations Suppose that (B-M)=>(M1) (d.B-M)=>(d.M1)
H. Groemer, On an inequality of Minkowski for mixed volumes, Geom. Dedicata. 33 (1990),
(B-M)=>(M1)
R.J.G. and S. Vassallo,, J. Math. Anal. Appl.
231 (1999), and 245 (2000),
(d.B-M)=>(d.M1)
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11. Projections and Sections
Kubota:
Dual Kubota:
section function
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12. Duality in Geometric Tomography
Convex bodies
Star bodies (?)
Projections
Sections through o
Support function
Radial function
Brightness function
Section function
Projection body
Intersection body
Cosine transform
Spherical Radon transform
Mixed volumes
Dual mixed volumes
Brunn-Minkowski ineq.
Dual B-M inequality
Aleksandrov-Fenchel
Dual A-F inequality
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13. Extending the Dual B-M Theory
General definition: If C is a bounded Borel subset of a k-dimensional subspace S, then
and
R.J.G., E. B. V. Jensen, and A. Volčič, Geometric tomography and local stereology, Adv. in Appl. Math. 30 (2003),
Extension of definition of
is problematic…
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14. Local Stereology Eva B. Vedel Jensen and H. J. Gundersen, c. 1985
based on an isotropic k-dimensional
subspace S is an unbiased estimator
of Vn(C).
For example, with n=3 and k=1, S
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15. Intersection Bodies Erwin Lutwak
Intersection bodies and dual mixed volumes,
Adv. in Math. 71 (1988),
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16. Busemann Intersection Inequality
If K is a convex body in En containing the origin in its interior, then
with equality if and only if K is an origin-symmetric ellipsoid.
H. Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1-12.
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17. Petty’s Conjectured Projection Inequality
C. Petty, 1971.
Let K be a convex body in En. Is it true that
with equality if and only if K is an ellipsoid?
Petty Projection Inequality:
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18. Generalized Busemann Intersection Inequality
If C is a bounded Borel set in En and
, then
with equality when if and only if C is an origin-symmetric ellipsoid and when
if and only if C is an origin-symmetric star body, modulo sets of measure zero.
H. Busemann and E. Straus, 1960; E. Grinberg, 1991; R. E. Pfiefer, 1990
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19. Dual Affine Quermassintegrals
Extended B.I.I.:
If C is a bounded Borel set in En and
, then
is the (n-i)th dual affine quermassintegral.
E. Grinberg, 1991
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20. Affine Quermassintegrals
If K is a convex body in En and , then
is the (n-i)th affine quermassintegral.
E. Lutwak, 1984; E. Grinberg, 1991
Lutwak:
R.J.G., 2005+:
for bounded Borel sets star-shaped at the origin.
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21. An Open Problem
Let and suppose that K is a convex body in En. Is it true that
with equality when if and only if K is an ellipsoid?
Conjectured by E. Lutwak, 1988.
Inequality true for quermassintegrals and harmonic quermassintegrals.
Case i=0, j=1 is the Petty Projection Inequality.
Case i=0, j=n-1 and K origin symmetric is the Blaschke-Santaló Inequality
for origin-symmetric K:
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22. A Dual Problem
Let and suppose that C is a bounded Borel set in En. Is it true that
with equality when if and only if C is an origin-symmetric ellipsoid?
As stated in R.J.G., Geometric Tomography, Problem 9.6.
Inequality true for dual quermassintegrals (R.J.G., 2005+).
Case i=0 is the Extended Busemann Intersection Inequality (after replacing j by n-i).
Equality condition should be modified appropriately.
Inequality is false in general when (R.J.G.,2005+).
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23. Why? Let Open problem: Extended B.I.I.:
But when C is an origin-symmetric star body that is not an ellipsoid, we have
and
for
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24. B.I.I., Further Extended If and then
R.J.G. (2005+), with thanks to Gaoyong Zhang. (Work in progress.)
Equality conditions fully determined.
Case q=i and j=p=n is the Extended Busemann Intersection Inequality.
Proof uses known “random simplex” inequalities, Blaschke-Petkantschin
formula, Jensen’s inequality for integrals.
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