Bonnesen-style Wulff isoperimetric inequality

The Wulff isoperimetric inequality is a natural extension of the classical isoperimetric inequality (Green and Osher in Asian J. Math. 3:659-676 1999). In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities.

K of area A K and perimeter L K , by growing in the unit-speed along the direction of the outward normal, the area of the corresponding domain, which is denoted by A K (t), is a polynomial in t, which is known as the Steiner polynomial, that is, The discriminant of A K (t) =  is (K) is the isoperimetric deficit of K , which is non-negative by the following classical isoperimetric inequality: where the equality holds if and only if K is a disc. One can find some simplified and beautiful proofs of () that lead to generalizations to higher dimensions and applications to other branches of mathematics (cf. [-]). During the s, Bonnesen proved a series of inequalities of the form where B K is a non-negative geometric invariant and vanishes only when K is a disc. The inequality of the form () is known as the Bonnesen-style inequality, and the typical one was proved by Bonnesen himself (cf. [, ]): where R and r are the radius of the minimum circumscribed disc and the radius of the maximum inscribed disc of K , respectively. Many B K s were found in the last century, and mathematicians are still working on those unknown isoperimetric deficit lower limits of geometric significance. For more details, see references [, -].
If instead K grows by varying the outward normal speed to be a function p W (θ ) of the direction of the unit normal, one has the Wulff flow. The area of the domain when the initial domain K is convex and the function p W (θ ) is a support function of the convex body W with area A W is a polynomial in t in this flow, called the Wulff-Steiner polynomial, that is (cf. []), where L K,W is the Wulff length of ∂K with respect to W , and namely, where ∂K is the boundary of K , and s is the arc length parameter of ∂K . The discriminant of A K,W (t) =  is defined as the Wulff isoperimetric deficit (cf. []): When domain W is a unit disc, the Wulff isoperimetric deficit W (K) is the isoperimetric deficit of K . Let If W is the unit disc, then r W and R W are, respectively, the radius of the maximum inscribed disc and the radius of the minimum circumscribed disc of K .
We first prove the following Wulff isoperimetric inequality: where the equality holds if and only if K and W are homothetic. Then we consider the inequality of the form where B W (K) is an invariant of geometric significance of K and W and vanishes only when K and W are homothetic. The inequality of type () is called the Bonnesen-style Wulff isoperimetric inequality. Its reverse form, that is, Each equality holds if and only if K and W are homothetic.
Each equality holds if and only if K and W are homothetic.

Preliminaries
The support function p K (u) of the convex body K is defined by where u ∈ S  . For simplicity, we replace p K (u) by p K . For two convex bodies K, W , we have If the support functions of the convex bodies K, W are denoted by p K , p W , respectively, and t  , t  ∈ R, then the support function of The image of the convex body K at time t ≥  under the normal flow having speed p W (u) (the Wulff flow associated to W ) is K + tW .
Let p K be the support function of K , then

Proposition  (Poincaré lemma []) Let f be a function on [, a] whose first derivative is square integrable and such that
where equality holds if and only if f = A cos(πx/a) + B sin(πx/a). In particular, if a ≤ π , then

Inequality () holds as an equality if and only if
for some constant c.

Bonnesen-style Wulff isoperimetric inequalities
To prove our main results, we need the following lemmas.
, since the angle between u and v is strictly greater than π  , hence u · v < . By (), we have For >  small enough, we have Hence Lemma  Let K, W be two oval bodies in R  . Let r W , R W be, respectively, the W-inradius and W-outradius of K . Then the equation A K,W (t) =  has two roots t  , t  such that Each inequality in () holds as an equality if and only if K and W are homothetic. In particular, when r W ≤ t ≤ R W , Inequality () is strict whenever r W < t < R W . When t = r W or t = R W , equality will occur in () if and only if K and W are homothetic.
Proof There is at least one point where ∂(r W W ) is tangent to ∂K for θ ∈ [θ  , θ  + π] with all θ  . If the conclusion fails, that is, there exists θ  such that p r W W (θ ) = p K (θ ) for θ ∈ [θ  , θ  + π], choose the vector v corresponding to the angle θ  + π  . By Lemma , if we move r W W by v for >  small enough, then r W W + · v continues to lie in the interior of K and has no points of tangency. This contradicts the maximality of r W .
By integration by parts we have Let θ  , θ  , . . . , θ N be points where ∂(r W W ) are tangent to ∂K . We can break up the righthand side of () into integrals over the intervals [θ i , θ i+ ] ( ≤ i ≤ N -). Since every set [θ , θ + π] contains a point where r W W is tangent to K , we have at each point of tangency. Applying inequality () in the Poincaré lemma, we have where the equality holds if and only if Since the convex body K contains tW , then for all θ . This leads to c = , that is, K and W are homothetic. In a similar way, we have where the equality holds if and only if K and W are homothetic. Thus the equation A K,W (t) =  has two roots t  , t  , and and therefore In particular, according to (), when r W ≤ t ≤ R W , we have If r W < t < R W , then t  < -t < t  . Inequality () is strict. Therefore, equality occurs in () only when t = r W or t = R W , that is, K and W are homothetic. Lemma  is proved. By () or (), the sufficient condition for root existence of equation A K,W (t) =  is that the discriminant of A K,W (t) =  is non-negative. We obtain the following Wulff isoperimetric inequality.

the equality holds if and only if K and W are homothetic.
Proof of Theorem  By inequalities (), (), we have, respectively, Then inequalities (), () can be, respectively, rewritten as Therefore, we have where the equality holds if and only if the equalities of (), () hold, that is, K and W are homothetic. This proves inequality (). Inequalities (), () can also be rewritten, respectively, as follows: Hence, we have where the equality holds if and only if K and W are homothetic. Inequality () is proved.
Let W be the unit disc, then L  K,W = L  K , A W = π . Therefore we have the following.
Corollary  Let K be an oval body in R  with area A K and perimeter L K . Let r and R be, respectively, the radius of the maximum inscribed disc and the radius of the minimum circumscribed disc of K . Then

Each equality holds if and only if K is a disc.
It should be noted that () is obtained in [], which is stronger than the Bonnesen isoperimetric inequality ().

Reverse Bonnesen-style Wulff isoperimetric inequalities
To prove reverse Bonnesen-style Wulff isoperimetric inequalities in Theorem , we need the following Wulff isoperimetric inequalities.
Lemma  Let K, W be two oval bodies in R  with areas A K and A W . Let r W , R W be, respectively, the W-inradius and W-outradius of K . Then Each equality holds if and only if K and W are homothetic.
Proof The Wulff isoperimetric inequality can be rewritten as Each inequality holds as an equality if and only if K and W are homothetic. Recalling (), () and (), we have By the definition of r W , we have for all θ , which leads to where the equality holds if and only if r W p W = p K for all θ , that is, K and W are homothetic. By the definition of L K,W in (), we have Via the area formula (), we have Hence, we have where the equality holds if and only if R W p W = p K for all θ , that is, K and W are homothetic. By (), () and (), we have Inequalities () are proved. Inequalities (), () can, respectively, be rewritten as Together with () and the above inequalities, inequalities () follow.
Proof of Theorem  By inequalities (), we have where the equality holds if and only if each equality of () holds, that is, K and W are homothetic. This is inequality (). By inequalities (), we have where the equality holds if and only if each equality of () holds, then K and W are homothetic. This is inequality ( From the equality conditions of () and () again, the equality of () holds if and only if K and W are homothetic. This gives inequality (). Theorem  is proved.
Let W be a unit disc. Direct consequences of Theorem  are as follows.
Corollary  Let K be an oval body in R  with area A K and perimeter L K . Let r and R be, respectively, the radius of the maximum inscribed disc and the radius of the minimum circumscribed disc of K . Then

Each equality holds if and only if K is a disc.
The reverse Bonnesen-style inequality () is obtained by Bokowski, Heil, Zhou, Ma and Xu (cf. [, ]).