Narrowing the gaps of the missing Blaschke-Santaló diagrams

In 1916 Blaschke [3] raised the question of determining the set of triples given by the volume $V(K)$, surface area $S(K)$, and integral mean curvature $M(K)$ of every set $K$ from the set of $3$-dimensional convex bodies $\mathcal{K}^{3}$. To do so, he defined the function

and studied the set of images $h(\mathcal{K}^{3})$ of $h$, which is known as the Blaschke diagram. Despite recent efforts (see [29] or [33]) this diagram still today remains unsolved.

Years later, Santaló [34] proposed the study of possible triples for the geometric quantities area $A$, perimeter $p$, inradius $r$, circumradius $R$, diameter $D$, and minimum width $w$ of a $2$-dimensional convex body. As a result, he already solved the cases $(A,p,w)$, $(A,p,r)$, $(A,p,R)$, $(A,D,w)$, $(p,D,w)$, and $(R,r,D)$. Later on Hernández Cifre and Segura Gomis in [19] and [23] concluded the quest to describe the diagrams $(R,r,w)$, $(R,D,w)$, $(r,D,w)$, Hernández Cifre completely described in [20] the diagrams $(A,D,R)$ and $(p,D,R)$, Böröczky Jr., Hernández Cifre, and Salinas described the diagrams $(A,r,R)$ and $(p,r,R)$ in [5], and very recently Delyon, Henrot, and Privat completely described in [12] the diagram $(A,r,D)$.

Those functionals have been also considered in other diagrams. For instance, it has been described the case $(R,r,D,w)$ in [6] as well as an almost complete description by Ting and Keller of $(A,p,D,w)$ in [38]. Furthermore, Hernández Cifre et al. described in [21] diagrams for the inradius, circumradius, and volume and surface area for $n$-dimensional centrally symmetric convex bodies. In [7] the authors described detailed $(R,r,D)$ diagrams when the functionals are measured with respect to other balls different from the Euclidean ball, such as the triangle, square, pentagon, or hexagon, or a general convex body. Very recently in [10] the authors studied $(R,r,D)$ diagrams for other notions of diameters.

More diagrams have been recently studied when introducing other functionals. Ftouhi and Lamboley [16] have described the diagram for the area, perimeter, and the first Dirichlet eigenvalue, Lucardesi and Zucco [28] described the diagram for the area, the torsional ridigity, and the first Dirichlet eigenvalue (see also Buttazzo and Pratelli [11] for more information on this diagram), Ftouhi and Henrot [15] studied the diagram for the area, the first Dirichtel eigenvalue, and the first non-trivial Laplacian-Neumann eigenvalue, and also Ftouhi [14] computed the diagram of the area, perimeter, and the Cheeger constant. Very recently Gastaldello, Henrot, and Lucardesi in [17] have studied the diagram of area, perimeter, and moment of inertia.

The aim of this paper is to solve several new inequalities relating three of the six functionals proposed by Santaló, and to clarify what is known up to date in the matter. This is why we state and prove each new result in the corresponding section. Indeed, we devote a section to each of the six unsolved diagrams, with the exception of $(A,r,w)$ and $(p,r,w)$ which use a common one.

Let $\mathcal{K}^{n}$ be the set of $n$-dimensional convex bodies, i.e., convex and compact sets in $\mathbb{R}^{n}$ different from points.

For every $x,y\in\mathbb{R}^{n}$, $x=(x_{1},x_{2},\dots,x_{n})$, $y=(y_{1},y_{2},\dots,y_{n})$, let $\langle x,y\rangle:=x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}$ be the scalar product of $x$ and $y$. Moreover, let $\|x\|:=\sqrt{\langle x,x\rangle}$ be the Euclidean norm of $x$. Furthermore, let $e_{1},e_{2},\dots,e_{n}$ be the vectors of the canonical base of $\mathbb{R}^{n}$.

Let $K\in\mathcal{K}^{n}$. The circumradius (resp. inradius) $R(K)$ (resp. $r(K)$) of $K$ is the smallest (resp. largest) radius of an Euclidean ball containing $K$ (resp. contained in $K$). The diameter $D(K)$ of $K$ is the largest Euclidean length between two points contained in $K$. The (minimum) width is the minimum distance between two parallel hyperplanes containing $K$ between them. For every $K\in\mathcal{K}^{2}$, let $p(K)$ be the perimeter of $K$. For every $K\in\mathcal{K}^{n}$, let $V(K)$ be its volume or Lebesgue measure, and if $n=2$, realize that $V$ coincides with the usual area $A$.

These functionals are monotonically increasing with respect to inclusion, i.e., if $K,C\in\mathcal{K}^{n}$, then $f(K)\leq f(C)$ for $f\in\{R,r,D,w,V\}$ and if $n=2$ then $f(K)\leq f(C)$ for $f=p$ too. Moreover, if $f\in\{p,R,r,D,w\}$ then $f$ is $1$-homogeneous, i.e. $f(\lambda K)=\lambda f(K)$ and $\lambda\geq 0$. In the case of the volume, it is $n$-homogenous, i.e. $V(\lambda K)=\lambda^{n}V(K)$, for every $K\in\mathcal{K}^{n}$ and $\lambda\geq 0$, and thus $A$ is $2$-homogeneous.

Some remarkable examples of planar convex bodies are the Euclidean unit ball $\mathbb{B}$, the equilateral triangle of unit circumradius $\mathbb{T}$, the line segment of edge length $2$ $\mathbb{L}$, and the Reuleaux triangle of unit circumradius $\mathbb{RT}$. A body $K\in\mathcal{K}^{2}$ is of constant width if $w(K)=D(K)$ (see [30] for further details). In $n$-dimensional space $\mathbb{R}^{n}$, let $\mathbb{B}_{n}$ be the $n$-dimensional Euclidean unit ball, thus having that $\mathbb{B}_{2}=\mathbb{B}$.

Since this is eminently a paper devoted to planar sets, when we refer to some result, we will directly specify its $2$-dimensional version. Moreover, for the references to most of the known inequalities we will refer to [1] as well as the references therein. However, we state here the inequalities that we need further on.

Let $K\in\mathcal{K}^{2}$. The isoperimetric inequality (see [31]) states that

with equality if and only if $K=\mathbb{B}$.

Pál’s inequality (see [32]) provides a lower bound of $A$ in terms of $w$, namely

with equality if and only if $K=\mathbb{T}$. If $K$ is moreover assumed to be of constant width, then the inequality strengthens onto

known as the Blaschke-Lebesgue inequality (see [4] or [27]), with equality if and only if $K=\mathbb{RT}$.

Further trivial inequalities between the area and the circumradius and inradius are

with equality if and only if $K=\mathbb{B}$.

The perimeter and the diameter relate as

where the left inequality is trivial and gets equal when $K=\mathbb{L}$, and the right one is a consequence of Barbier [2] and it becomes equality when $K$ is a constant width set.

The perimeter and the inradius fulfill the trivial inequality $2\pi r(K)\leq p(K)$ with equality for $K=\mathbb{B}$. The perimeter and the circumradius fulfill

where the left inequality attains equality when $K=\mathbb{L}$, and the right inequality gets equal when $K=\mathbb{B}$.

The relation between the perimeter and the width is described by

as a consequence of Cauchy’s formula (see [35]), with equality if and only if $K$ is of constant width.

In [5] it was proven the optimal lower bound for the perimeter in terms of the inradius and the circumradius, namely:

Moreover, equality holds if and only if $K$ is the convex hull of a point, its mirrored with the origin, and an euclidean ball centered at the origin.

In [25] the author derived the optimal upper bound of the perimeter in terms of the diameter and the width

Moreover, equality holds for the intersection of the Euclidean ball centered at the origin with a halfspace and its mirrored with respect to the origin.

The quotient between the circumradius and the diameter is described by Jung’s inequality [24]

with equality if and only if $K$ contains an equilateral triangle of circumradius $R(K)$ and diameter $D(K)$.

The relation between the inradius and the width is described by

where the left side attains equality if and only if $K=\mathbb{B}$ and the right side, known as Steinhagen’s inequality [37], becomes equality if and only if $K=\mathbb{T}$.

The relation between the circumradius and the width is described by

with equality if and only if $K=\mathbb{B}$.

Finally, if $K$ is of constant width, it is known that $2R(K)\leq(\sqrt{3}+1)r(K)$ with equality if and only if $K=\mathbb{RT}$. This last statement is a consequence of (9) and the fact that constant width sets attain equality in the concentricity inequalities

(see [8] for more information on these inequalities). Moreover, precisely in the case of constant width sets, one can obtain the sharpened Steinhagen’s inequality

with equality if and only if $K=\mathbb{RT}$.

Let $K\in\mathcal{K}^{n}$. Given a hyperplane $L\subset\mathbb{R}^{n}$, we say that $K^{\prime}$ is the Steiner symmetrization of $K$ with respect to $L$ if for every $x\in L$, $(x+L^{\bot})\cap K^{\prime}$ is a segment centered around $x$, and it has the same length than $(x+L^{\bot})\cap K$. It is well known that $D(K^{\prime})\leq D(K)$, $R(K^{\prime})\leq R(K)$, $r(K)\leq r(K^{\prime})$, $V(K^{\prime})=V(K)$, and if $n=2$ then $A(K^{\prime})=A(K)$ and $p(K^{\prime})\leq p(K)$ (see [35]).

For every $X\subset\mathbb{R}^{n}$, let $\mathrm{aff}(X)$, $\mathrm{lin}(X)$, and $\mathrm{conv}(X)$ be the affine, linear, and convex hull of $X$, respectively. Given $x,y\in\mathbb{R}^{n}$, we let $[x,y]=\mathrm{conv}(\{x,y\})$ be the line segment with endpoints $x$ and $y$. Let $\partial K$ to be the boundary of $K$.

The circumball is characterized by some touching conditions of boundary points (see for instance [9]).

An analogous result applies to the inradius, namely, if $\mathbb{B}_{n}\subset K$, then $r(K)=1$ if and only if there exist $x_{i}\in\partial K\cap\partial\mathbb{B}_{n}$, $i=1,\dots,j$, $j\in\{2,\dots,n+1\}$, such that $0\in\mathrm{conv}(\{x_{1},\dots,x_{j}\})$. In particular, the points $x_{i}$ coincide with outer normals of $K$ and $\mathbb{B}_{n}$, and thus, whenever $n=2$ and $j=3$, the intersection of the halfspaces determined by the orthogonal lines to each $x_{i}$ containing $x_{i}$, $i=1,2,3$, determines a triangle $T$ with the property that

Let us consider the diagram $f_{ApD}(\mathcal{K}^{2})$ where

(see Figure 1).

We expose the best known inequalities relating the three quantities $A$, $p$, and $D$ for $K\in\mathcal{K}^{2}$.

First of all, notice that (4) induces a linear boundary of the diagram $f_{ApD}(\mathcal{K}^{2})$. Indeed, $K$ fulfills (4) with equality if and only if $K$ is of constant width. In particular, from (1) and (3) we know that if $K$ is of constant width with $w(K)=2$, then $2(\pi-\sqrt{3})\leq A(K)\leq 4\pi$, with $K=\mathbb{RT}$ and $K=\mathbb{B}$ attaining the lowest and highest values (see Figure 1).

Second, Kubota [25] proved that

where $\varphi$ is defined to be solution of $2\varphi D(K)=p(K)\sin\varphi$, with equality for $K$ being a symmetric lenses, i.e., the intersection of two Euclidean balls of the same diameter. It is worth mentioning that (14) implies a strengthening of the isoperimetric inequality (1) (see Figure 1¹¹1Here and on each other diagram, we represent valid inequalities by lines. In case of continuous lines, we refer to real boundaries of the diagram; in case of dashed lines, not. If we draw $K$ on a diagram, it coincides with the value $f(K)$ of its image through the corresponding diagram.).

Third, Kubota [25, 26] also proved that if $2D(K)\leq p(K)\leq 3D(K)$ then

with equality for isosceles triangles with two equal longer edges, having as extreme cases $K=\mathbb{L}$ and $K=\mathbb{T}$. Moreover, in the case of $3D(K)\leq p(K)\leq\pi D(K)$, he also proved the inequality

with equality only when $K=\mathbb{T}$.

Let us consider the diagram $f_{pDr}(\mathcal{K}^{2})$ where

(see Figure 2).

It is well known (see [18, (3)]) that

with equality for $K=\mathbb{L}$.

It is also known that the right-hand side of (4) induces part of the upper boundary of the diagram, where constant width sets attain equality, and with extreme points at $K=\mathbb{RT}$ (due to (12)) and $K=\mathbb{B}$.

We now establish a new inequality, providing the lower boundary of this diagram.

We now derive an easy non tight upper bound of the diagram as a consequence of (8).

Let us consider the diagram $f_{pRw}(\mathcal{K}^{2})$ where

(see Figure 3).

The inequality (6) induces part of the right boundary of the diagram. Remember that $K$ belongs to this boundary if and only if $K$ is of constant width. One of the extreme points is $K=\mathbb{B}$. On the other hand, if $K$ is of constant width, i.e. $D(K)=w(K)$, together with (9), implies that $\sqrt{3}R(K)\leq w(K)$, with equality when $K=\mathbb{RT}$. Thus, this last set is the other extreme point of this boundary.

We now show a new inequality which induces the upper boundary of the diagram.

We can easily derived lower bounds, for instance, using (10) in combination with (7).

Let us consider the diagram $f_{ARw}(\mathcal{K}^{2})$ where

(see Figure 4).

Apparently, the only two known inequalities relating this triple are

(see [18]) with both inequalities having equality if $K=\mathbb{L}$, and the left-hand side also attaining equality at $K=\mathbb{T}$.

The first inequality we prove induces the upper boundary of the diagram.