The problem about a symmetric convex body that is lifted from shallow water

For a convex body $ K $ in $ {\mathbb R}^{n} $, Lutwak, Yang and Zhang defined a new ellipsoid $ \Gamma_{-2}K $, which is the dual analog of the Legendre ellipsoid. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $ K $, there exist an ellipsoid $ E $ and a parallelotope $ P $ such that $ \Gamma_{-2}E \supseteq \Gamma_{-2}K \supseteq \Gamma_{-2}P $ and $ V(E)=V(K)=V(P) $; (ii) For any convex body $K$ whose John point is at the origin, then there exists a simplex $T$ such that $ \Gamma_{-2}K \supseteq \Gamma_{-2}T $ and $ V(K)=V(T) $.

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On the Local Convexity of Intersection Bodies of Revolution

Canadian Journal of Mathematics ◽

10.4153/cjm-2013-039-4 ◽

2015 ◽

Vol 67 (1) ◽

pp. 3-27

Author(s):

M. Angeles Alfonseca ◽

Jaegil Kim

Keyword(s):

Convex Body ◽

Convex Geometry ◽

Euclidean Ball ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Body Of Revolution ◽

Local Convexity ◽

Bodies Of Revolution ◽

Intersection Body ◽

Intersection Bodies

AbstractOne of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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Almost ellipsoidal sections and projections of convex bodies

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051355 ◽

1975 ◽

Vol 77 (3) ◽

pp. 529-546 ◽

Cited By ~ 5

Author(s):

D. G. Larman ◽

P. Mani

Keyword(s):

Convex Body ◽

High Dimension ◽

Interior Point ◽

Convex Bodies ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Centrally Symmetric ◽

Dimensional Section ◽

Projections Of Convex Bodies

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

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Two finiteness theorems in the Minkowski theory of reduction

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010818 ◽

1972 ◽

Vol 14 (3) ◽

pp. 336-351 ◽

Cited By ~ 2

Author(s):

P. W. Aitchison

Keyword(s):

Convex Body ◽

Euclidean Space ◽

Quadratic Forms ◽

Positive Definite ◽

Dimensional Euclidean Space ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Reduction Theory ◽

Positive Definite Quadratic Form ◽

Finiteness Theorems

Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

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On the volume product of planar polar convex bodies — Lower estimates with stability

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.2.1235 ◽

2013 ◽

Vol 50 (2) ◽

pp. 159-198

Author(s):

K. Böröczky ◽

E. Makai ◽

M. Meyer ◽

S. Reisner

Keyword(s):

Convex Body ◽

Rotational Symmetry ◽

Polar Body ◽

Convex Bodies ◽

Stability Estimate ◽

The Body ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

The Stability ◽

Area Product

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(K − K)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

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On the Inequality for Volume and Minkowskian Thickness

Canadian Mathematical Bulletin ◽

10.4153/cmb-2006-019-4 ◽

2006 ◽

Vol 49 (2) ◽

pp. 185-195 ◽

Cited By ~ 4

Author(s):

Gennadiy Averkov

Keyword(s):

Convex Body ◽

Geometric Inequality ◽

Symmetric Convex Body ◽

Symmetric Convex ◽

Centrally Symmetric ◽

Arbitrary Convex ◽

Simple Corollary ◽

The Difference ◽

The Right ◽

Finite Dimensional Banach Space

AbstractGiven a centrally symmetric convex body B in , we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ball B. Let K be an arbitrary convex body in ℳd(B). The relationship between volume V(K) and the Minkowskian thickness (= minimal width) ΔB(K) of K can naturally be given by the sharp geometric inequality V(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain that with equality on the left attained if and only if B is the difference body of a simplex and on the right if B is a cross-polytope. The main result of this paper is that for d = 2 the equality on the right implies that B is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

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