On some discrete Bonnesen-style isoperimetric inequalities

The isoperimetric problem is an ancient problem in geometry that has a significant impact on various branches of mathematics. The problem deals with finding a closed curve or surface that encloses the maximum area or volume for a given perimeter or boundary length. The geometric inequalities derived from the isoperimetric problem have far-reaching effects and connections to other areas of mathematics. In the 1950s, a connection was discovered between the isoperimetric problem and the Sobolev embedding problem. The link between the two problems provided insights into the relationship between geometric and functional analysis. In the 1970s, the Aleksandrov-Fenchel inequality, which is a generalization of the isoperimetric inequality, was found to be closely related to the Hodge index theorem in algebraic geometry. In recent decades, there has been extensive research on convex geometry related to the isoperimetric problem. This research has established important connections among various fields, including functional analysis, harmonic analysis, affine geometry, partial differential equations, and information theory.

The complete mathematical proof of isoperimetric inequality was not established until the variational method based on calculus appeared in the 19th century. After this breakthrough, various proofs for the isoperimetric inequality were developed. In 1962, Hurwitz proposed two ingenious methods to prove the isoperimetric inequalities. He employed Fourier analysis in his proofs, one for convex curves and the other for general curves. These methods were detailed in his works [3, Section 4.2] and [5, Page 392-394], respectively. The planar isoperimetric inequality can be derived as an immediate consequence of Poincaré formula [1, Section 23], which can be founded in references such as [4, Chapter 7, Section 7] and [8, Page 1183-1185]. Zhang demonstrated that the Sobolev inequality, a key result in functional analysis, is a special form of the isoperimetric inequality ([20].Zhou utilized the fundamental kinematic formula to establish Bonnesen-type isoperimetric inequalities and other new inequalities in integral geometry ([10, 21, 22]). It is worth mentioning that Zhang [17] introduced the discrete Wirtinger inequality to obtain some new analytical isoperimetric inequalities.

The description of classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^{2}$ is: assume that $K$ is a domain with length $L$ and area $A$, then

where the equality holds if and only if $K$ is a disc.

In 1920, Bonnesen discovered a series of inequalities in $\mathbb{R}^{2}$ as follows ([10, 8, 9, 15])

where $B_{k}$ is non-negative with geometric significance and vanishes only when $K$ is a disc.

Denote by $\Delta\left(K\right)=L^{2}-4\pi A$ the isoperimetric deficit of $K,$ and $B_{K}$ measures the “deviation” between $K$ and a disc. Many $B_{K}$’s for planar domains are found by mathematicians. For instance, Bonnesen, Zhang, Zhou etc. obtained a series of Bonnesen-type isoperimetric inequalities ([8, 9, 16]): assume that $K$ is the domain of length $L$ and area $A.$ Let $r_{K}$ and $R_{K}$ be the maximum inscribed radius and minimum circumscribed radius, respectively. Then

and

With equalities hold when and only when $K$ is a disc.

Comparing these $B_{K}$’s and determining the best lower bound poses a challenge, and mathematicians are still pursuing to discover these unknown invariants of geometrical significance.

Rencently, mathematicians have turned their attention towards the discrete isoperimetric problem, such as polygons or polyhedrons. In fact, by establishing a series of analytic inequalities that related to solutions of nonlinear second-order differential equalities, Zhang [17, Section 3] obtained some discrete isoperimetric inequalities of polygons in $\mathbb{R}^{2}$.

Assume that $H_{n}$ is an $n$-sided planar convex polygon. Denote by $L_{n}$ and $A_{n}$ length and area of $H_{n}$, respectively. Then

The quantity $L_{n}^{2}-4d_{n}A_{n}$ is called the isoperimetric deficit of $H_{n}$. The sharp discrete Bonnesen-style isoperimetric inequality has the following form

where $U_{n}$ is non-negative with geometric significance and vanishes only when $H_{n}$ is a regular polygon. $U_{n}$ measures the “deviation” of $H_{n}$ from “regularity”.

Utilizing analytic inequalities such as the discrete Wirtinger inequality and Schur-convex (concave) function have been proved to be an effective method for solving discrete isoperimetric problem (see [13, 17, 18, 19]). However, the discrete Bonnesen-style isoperimetric inequalities remain largely unexplored, with only a few inequalities discovered in $\mathbb{R}^{2}$. As noted in Zhang’s paper [17], a convex polygon that is inscribed in a circle is called as “cyclic” and such polygons always enclose the largest area. Therefore, in studying geometric inequalities for planar convex polygon, one only needs to consider cyclic polygons.

Assume that ${\Lambda}_{n}$ is an $n$-sided planar convex polygon with perimeter $L_{n}$ and area $A_{n}$, which is inscribed in a circle with radius $R$. Meanwhile, there exist a regular polygon with perimeter $L_{n}^{\ast}$ and area $A_{n}^{\ast}$ inscribed in the same circle ${\Lambda}_{n}$. Zhang [18] achieved the following geometric inequality for planar convex polygon in $\mathbb{R}^{2}$

where equality holds when and only when $\Lambda_{n}$ is a regular polygon.

In this paper, inspired by Zhang, Qi and Ma’s work ([19, 13]), by establishing two analytic isoperimetric inequalities (Theorem 1, 2), we derive a series of discrete Bonnesen-style isoperimetric inequalities (Theorem 3, 4) and their reverse forms (Theorem 5, 6). In Section five, we make up for the shortcomings in the proof of Ma’s theorem in [6], and present a new proof of Zhang and Ma’s results ([18, 6])(Theorem 7). Meanwhile, we suppose that on surfaces of constant curvature, Schur convexity also will be a convenient and excellent idea to explore geometric inequalities on surfaces of constant curvature.

The Schur-convex (concave) function was introduced by Schur in 1923 ([11]) and has many important applications in analytic inequalities. In this section, we review some basic facts about Schur-convex (concave) function and give two important analytic inequalities.

An $n\times n$ matrix $P=\left[p_{ij}\right]$ is called a doubly stochastic matrix if $p_{ij}\geq 0$ for $1\leq i$, $j\leq n$, and

For instance, $P=\left[p_{ij}\right]$ with $p_{ij}=\frac{1}{n}$, $1\leq i$, $j\leq n$ is a doubly stochastic matrix.

Let $I$ be an open interval of the real number line R, and $I^{n}=I\times I\times\cdots\times I$. A real function $f:I^{n}\rightarrow\textbf{R}$ ($n\geq 2$) is Schur-convex if for any doubly stochastic matrix $P$ and $x\in I^{n}$, then

Furthermore, $f$ is called strictly Schur-convex if (2.1) is strict. Similarly, $f$ is strictly Schur-concave if (2.1) is inverse and strict.

A real function $f:I^{n}\rightarrow\textbf{R}$ ($n\geq 2$) is symmetric if for any permutation matrix $T$ and $x\in I^{n}$

Every Schur-convex function is symmetric, but not every symmetric function can be a Schur-convex function. The following lemma is defined as the Schur’s condition.

Due to the symmetry of $f(x)$, the Schur’s condition (2.3) can be rewritten as ([7, Page 57])

and $f$ is strictly Schur-convex if (2.4) is strict for $x_{1}\neq x_{2}$.

The following notations will be often used in the latter part of this article.

We obtain the following analytic inequality.

In particular, if $f\left(\theta\right)=\tan{\theta}$ and $f\left(\theta\right)=\sec{\theta}$ for $\theta\in\left(0,\frac{\pi}{2}\right)$, then

Notice that $f\left(x\right)=x^{2}$ on $\left(0,1\right)$ has strict convexity, then

In particular, assume that $f\left(\theta\right)=\tan{\theta}$ and $f\left(\theta\right)=\csc{\theta}$ for $\theta\in\left(0,\frac{\pi}{2}\right)$, then we have

In this section, by using the analytic isoperimetric inequalities in Theorem 1, we establish some discrete Bonnesen isoperimetric inequalities.

Assume that $\alpha=1$, we have the following special case.

Similarly, substituting (3.3) and (3.4) into (2.13), we obtain

(3.8) can be regard as the reverse inequality of (3.7). When $\alpha=1$, we have

It is interesting and also difficult to study the reverse Bonnesen-style isoperimetric inequalities. There are only a few reverse Bonnesen-style isoperimetric inequalities of convex domains (see [2]). In 1993, Bottema discovered a classic reverse Bonnesen-style isoperimetric inequality of oval domains in $\mathbb{R}^{2}$([10]). The discrete cases are more complex and difficult.

Assume that $K$ is a convex domain with $C^{2}$ smooth boundary in $\mathbb{R}^{2}$, then ([10])

where $\rho_{M}$ and $\rho_{m}$ are the maximum and minimum value of $\rho\left(\partial K\right)$, respectively. The equality holds when and only when $\partial K$ is a circle.

Mathematicians still continue to work hard on investigating the inverse discrete Bonnesen-type isoperimetric inequalities. A natural question is that: for an $n$-sided planar convex polygon if there is a geometric invariant $D_{n}$ such that

where $D_{n}$ satisfies

1. (1)

   $D_{n}\geq 0$;

2. (2)

   $D_{n}=0$ only when $H_{n}$ is regular.

In order to obtain the inverse discrete Bonnesen-style isoperimetric inequalities, we use the analysis isoperimetric inequalities in Theroem 2. Substituting (3.3) and (3.4) into (2.22), then we have