On containment measure and the mixed isoperimetric inequality
© Xu et al.; licensee Springer. 2013
Received: 19 December 2012
Accepted: 30 August 2013
Published: 19 November 2013
We first investigate whether for given convex domains , in the Euclidean plane, for any rotation α, there is a translation x so that or . Then, we estimate the mixed isoperimetric deficit of domains and via the known kinematic formulas of Poincaré and Blaschke in integral geometry. We obtain the sufficient condition for domain to contain, or to be contained in, convex domain . Finally, we obtain the mixed isoperimetric inequality and some Bonnesen-style mixed inequalities. Those Bonnesen-style mixed inequalities obtained are the known Bonnesen-style inequalities if one of the domains is a disc. As a direct consequence, we obtain the strengthened Bonnesen isoperimetric inequality.
1 Introductions and preliminaries
A homothety of a convex set K is of the form for , . A convex body is a compact convex set with nonempty interior. A domain is a set with nonempty interior.
One may be interested in the following strong containment problem: Whether for given convex domains and , there exists a translation x so that or for any rotation α. It should be noted that this containment problem is much stronger than Hadwiger’s one. Therefore, the strong containment problem could lead to general and fundamental geometric inequalities (cf. [1–9]).
The well-known classical isoperimetric problem says that the disc encloses the maximum area among all domains of fixed perimeters in the Euclidean plane .
The equality sign holds if and only if Γ is a circle.
Its analytic proofs root back to centuries ago. One can find some simplified and beautiful proofs that lead to generalizations of the discrete case, higher dimensions, the surface of constant curvature and applications to other branches of mathematics (cf. [1, 3–5, 10–53]).
measures the difference between domain K of area A and perimeter P, and a disc of radius .
is vanish only when K is a disc.
Many s are found during the past. The main interest is still focusing on those unknown invariants of geometric significance. See references [3–5, 12, 17, 23, 31, 32, 36] for more details. The following Bonnesen’s isoperimetric inequality is well known.
where the equality holds if and only if K is a disc.
Since for any domain K in , its convex hull increases the area and decreases the perimeter , that is, and , then we have , that is, . Therefore, the isoperimetric inequality and the Bonnesen-style inequality are valid for all domains in if these inequalities are valid for convex domains.
In this paper, we first investigate the stronger containment problem: Whether for given convex bodies , in the Euclidean plane , there is a translation x so that or for any rotation α. Then we investigate the mixed isoperimetric deficit of domains and .
Therefore, we can only consider the convex domains when we estimate the mixed isoperimetric deficit low bound.
Via the kinematic formulas of Poincaré and Blaschke in integral geometry, we obtain sufficient conditions for convex domain to contain, or to be contained in, another convex domain for a translation x and any rotation α (Theorem 1 and Theorem 2). We obtain the mixed isoperimetric inequality and some Bonnesen-style mixed inequalities (Theorem 3, Theorem 4, Corollary 2, Corollary 3, Corollary 4, Theorem 5 and Theorem 6). One immediate consequence of our results is the strengthening Bonnesen isoperimetric inequality (Corollary 3). These new Bonnesen-style mixed inequalities obtained are fundamental and generalize some known Bonnesen-style inequalities (Corollary 5).
2 The containment measure
where denotes the number of points of intersection .
We consider the homothetic copy () of .
Moreover, if , then contains .
The isoperimetric inequality guarantees that . We complete the proof of the theorem. □
3 Bonnesen-style mixed inequalities
Let , the maximum inscribed radius of with respect to , and , the minimum circum scribed radius of with respect to . Note that , are, respectively, the maximum inscribed radius, the minimum circum radius of when is the unit disc. It is obvious that . Therefore, for neither contains nor it is contained in . Then by Theorem 1, we have the following.
we obtain the following.
where the equality holds if and only if , that is, and are discs.
We complete the proof of Theorem 4. □
Corollary 2 (Kotlyar)
where the equality holds if and only if both and are discs.
Let be the unit disc, then Theorem 4 immediately leads to the following inequality that strengthens the Bonnesen isoperimetric inequality (4).
where the equality holds if and only if K is a disc.
where the equality holds if and only if and are discs.
where is an invariant of and . is, of course, assumed to be nonnegative and vanishes only when both and are discs.
Therefore, we obtain the following Bonnesen-style mixed inequalities.
Each inequality holds as an equality if and only if both and are discs.
The Bonnesen function attains minimum value at . The Bonnesen quadratic trinomial has only one root when . This means that both and are discs. This immediately leads to the following results.
Each equality holds if and only if and are discs.
Each equality holds if and only if K is a disc.
It should be noted that the first inequality in (27) is due to Bonnesen, and he only derived some inequalities for 2-dimensional case and never had any progress for higher dimensions or 2-dimensional surface of constant curvature. One would be interested in the situations in higher dimensional space and in the surface of constant curvature. Related development in those areas can be found in [26, 35, 37, 63–66] and . More details for the isoperimetric inequality and Bonnesen style inequalities can be found in [67–80].
Authors would like to thank two anonymous referees for many helpful comments and suggestions that directly lead to the improvement of the original manuscript. The corresponding author is supported in part by the NSFC (No. 11271302) and the Ph.D. Program of Higher Education Research Fund (No. 2012182110020).
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