In:
Studia Scientiarum Mathematicarum Hungarica, Akademiai Kiado Zrt., Vol. 50, No. 2 ( 2013-06-1), p. 159-198
Abstract:
Let K ⊂ ℝ 2 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 ⊂ ℝ 2 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 ⊂ ℝ 2 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 *| ≧ n 2 sin 2 ( 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 *| ≧ n 2 sin 2 ( 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.
Type of Medium:
Online Resource
ISSN:
0081-6906
,
1588-2896
DOI:
10.1556/sscmath.50.2013.2.1235
Language:
Unknown
Publisher:
Akademiai Kiado Zrt.
Publication Date:
2013
SSG:
17,1
Bookmarklink