Integral representations of quermassintegrals and Bonnesen-style inequalities

Bokowski, Jürgen; Heil, Erhard

doi:10.1007/bf01202503

Cited by 36 publications

(20 citation statements)

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“…The extensions given here involve volume, surface area, terms with integrals of certain mixed volumes, quermassintegrals of inner parallel bodies, and the inradius. The Wills conjecture has been shown to be false for n > 2 when the inradius is replaced with the circumradius; however, Bokowski and Heil [3] obtained a similar result with the inequality reversed. Chakerian [6] derived a Bonnesen-style inequality which yields a strengthened form of the isoperimetric inequality in two dimensions, but which does not involve volume and surface area in higher dimensions.…”

Section: Corollarymentioning

confidence: 77%

The Wills conjecture

Brannen¹

1997

Trans. Amer. Math. Soc.

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Abstract. Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to n-dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.One of the best known geometric inequalities is the isoperimetric inequality, which states that of all closed planar curves with fixed perimeter, a circle encloses the greatest area. If K is a planar convex body (a convex "body" is a compact convex set with nonempty interior) of perimeter L with area A, then this inequality can be expressed aswith equality only when the boundary of K is a circle.A strengthening of (1) was given by the Danish mathematician T. Bonnesen, who demonstrated in 1929 (see [5]) that if K has circumradius R and inradius r, thenThis inequality can be derived from the inequalityThe inequality (3) with λ = r is known as Bonnesen's inradius inequality.J. M. Wills [15] conjectured in 1970 thatwhich would be an extension of Bonnesen's inradius inequality to higher dimensions.

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Section: Corollarymentioning

confidence: 77%

The Wills conjecture

Brannen¹

1997

Trans. Amer. Math. Soc.

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“…In addition to establishing extensions of Bonnesen's inequality, bounds for the relative inradius and circumradius similar to those above will be discussed in the conclusion. The extension of Favard's inequality for n greater than 2 was obtained by Bokowski and Heil [4]. If we let R = R(K; B), i,j and k be integers such that 0 < i < j < k < n, and Cijk = (i + l)(k -j), then their inequality states…”

Section: ) Vviej) ~V(b)-[(-vw) -R^b))mentioning

confidence: 99%

“…The next inequalities involve polynomials whose coefficients are relative quermassintegrals as in (4). A discussion of Theorem (23) and its corollaries will precede the proof.…”

Section: T -Smentioning

confidence: 99%

Bonnesen-Style Inequalities for Minkowski Relative Geometry

Sangwine-Yager¹

1988

Transactions of the American Mathematical Society

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“…Many Bs are found in the last century and mathematicians are still working on those unknown invariants of geometric significance. See references [2], [3], [4], [6], [10], [17], [22], [25], [26], [39], [40], [47] for more details.…”

Section: Introductionmentioning

confidence: 99%