Some applications of means of convex bodies

Firey, William J.

doi:10.2140/pjm.1964.14.53

Cited by 20 publications

(24 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, the fact that JK and JL are ellipsoids in R n ensures that JK + 2 JL is also an ellipsoid. Indeed, Firey [10] proved this for n-dimensional ellipsoids. If JK or JL are not n-dimensional, choose sequences of n-dimensional ellipsoids E m and F m such that E m → JK and F m → JL as m → ∞ in the Hausdorff metric.…”

Section: A Characterization Of Minkowski Additionmentioning

confidence: 86%

A characterization of Blaschke addition

Gardner

Parapatits

Schuster

2014

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

Section: A Characterization Of Minkowski Additionmentioning

confidence: 86%

A characterization of Blaschke addition

Gardner

Parapatits

Schuster

2014

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…For convex bodies, harmonic p-combinations were investigated by Firey [9,10] (with applications presented in [11]). If Q # S n o and , # GL(n), then from the definition of the radial function it follows that \(,Q, x)= \(Q, , &1 x), for all x # R n .…”

Section: Dual Mixed Volumes and Firey Combinationsmentioning

confidence: 99%

The Brunn–Minkowski–Firey Theory II

Lutwak

1996

Advances in Mathematics

481

View full text Add to dashboard Cite

During the past two decades the notion of affine surface area (from affine differential geometry) and the isoperimetric inequalities related to it, have attracted increased interest. There are a number of reasons for this. First, there are new applications (see e.g. the survey of Gruber [12]). Then, there are the recently discovered extensions of affine surface area to arbitrary convex hypersurfaces (see e.g. Leichtwei? [15 18], Schu tt 6 Werner [37], Schu tt [36], Werner [38], and also [22]). These extensions have lead to recent verifications of the conjectured upper-semicontinuity and valuation property of classical as well as extended affine surface area (see [22,23,36]). Also, it has come to be recognized (see, e.g., [19,28,29]) that various isoperimetric inequalities involving affine surface area are very closely related to a variety of other important affine isoperimetric inequalities (e.g., the curvature image inequality, the Blaschke Santalo inequality, and Petty's geominimal surface area inequality).Geominimal surface area was introduced by Petty [30] more than two decades ago. As Petty stated, this concept serves as a bridge connecting affine differential geometry, relative differential geometry, and Minkowskian geometry. Both affine surface area and geominimal surface area are unimodular affine invariant functionals of convex hypersurfaces. Isoperimetric inequalities involving geominimal surface area are not only closely related to many isoperimetric inequalities involving affine surface area, but in fact, clarify the equality conditions of many of these inequalities.One of the aims of this article is to demonstrate that there are natural extensions of affine and geominimal surface areas in the Brunn Minkowski Firey theory. Surprisingly, it turns out that there are extensions of all of the known inequalities involving affine and geominimal surface areas to the new p-affine and p-geominimal surface areas. As will be article no. 0022 244

show abstract

“…We showed that numerous uniqueness results in Euclidean spaces are a consequence of a simple convexity property of the function that measures the ellipsoid's size. Most notably, minimal enclosing ellipsoids with respect to quermassintegrals are unique (see also [8,10]). …”

Section: Introductionmentioning

confidence: 99%

Minimal area conics in the elliptic plane

Weber

Schröcker

2012

Advances in Geometry

View full text Add to dashboard Cite

We prove uniqueness results for conies of minimal area that enclose a compact, fulldimensional subset of the elliptic plane. The minimal enclosing conic is unique if its center or axes are prescribed. Moreover, we provide sufficient conditions on the enclosed set that guarantee uniqueness without restrictions on the enclosing conies. Similar results are formulated for minimal enclosing conies of line sets as well.

show abstract

Some applications of means of convex bodies

Cited by 20 publications

References 6 publications

A characterization of Blaschke addition

A characterization of Blaschke addition

The Brunn–Minkowski–Firey Theory II

Minimal area conics in the elliptic plane

Contact Info

Product

Resources

About