1 Introduction and Preliminaries

Theorem 1.1

Let \(\mathcal {F}\) be a family of convex sets in \({\mathbb {R}}^{d}\) such that the volume of its intersection is \({\text {vol}}\left( \cap \mathcal {F}\right) >0\). Then there is a subfamily \(\mathcal {G}\) of \(\mathcal {F}\) with \(\left| \mathcal {G}\right| \le 2d\) and \({\text {vol}}\left( \cap \mathcal {G}\right) \le e^{d+1}d^{2d+\frac{1}{2}} {\text {vol}}\left( \cap \mathcal {F}\right) \).

We recall the note from [2] (see also [3]) that the number 2d is optimal, as shown by the 2d half-spaces supporting the facets of the cube.

The order of magnitude \(d^{cd}\) in the Theorem (and in the conjecture in [2]) is sharp as shown in Sect. 3.

Recently, other quantitative Helly type results have been obtained by De Loera et al. [5].

We introduce notations and tools that we will use in the proof. We denote the closed unit ball centered at the origin o in the d-dimensional Euclidean space \({\mathbb {R}}^{d}\) by \(\mathbf {B}\). For the scalar product of \(u,v\in {\mathbb {R}}^{d}\), we use \(\left\langle u,v\right\rangle \), and the length of u is \(|u|=\sqrt{\left\langle u,u\right\rangle }\). The tensor product \(u\otimes u\) is the rank one linear operator that maps any \(x\in {\mathbb {R}}^{d}\) to the vector \((u\otimes u)x=\left\langle u,x\right\rangle u\in {\mathbb {R}}^{d}\). For a set \(A\subset {\mathbb {R}}^{d}\), we denote its polar by \(A^{*}=\{y\in {\mathbb {R}}^{d}:\left\langle x,y\right\rangle \le 1 \text{ for } \text{ all } x\in A\}\). The volume of a set is denoted by \({\text {vol}}\left( \cdot \right) \).

Definition 1.2

We say that a set of vectors \(w_1,\ldots ,w_m\in {\mathbb {R}}^{d}\) with weights \(c_1,\ldots ,c_m>0\) form a John’s decomposition of the identity, if

$$\begin{aligned} \sum _{i=1}^{m} c_iw_i = o \;\; \text{ and } \;\; \sum _{i=1}^{m} c_i w_i\otimes w_i = I, \end{aligned}$$
(1)

where I is the identity operator on \({\mathbb {R}}^{d}\).

A convex body is a compact convex set in \({\mathbb {R}}^{d}\) with non-empty interior. We recall John’s theorem [8] (see also [1]).

Lemma 1.3

(John’s theorem) For any convex body K in \({\mathbb {R}}^{d}\), there is a unique ellipsoid of maximal volume in K. Furthermore, this ellipsoid is \(\mathbf {B}\) if, and only if, there are points \(w_1,\ldots ,w_m\in {\text {bd}}{\mathbf {B}}\cap {\text {bd}}{K}\) (called contact points) and corresponding weights \(c_1,\ldots ,c_m>0\) that form a John’s decomposition of the identity.

It is not difficult to see that if \(w_1,\ldots ,w_m\in {\text {bd}}{\mathbf {B}}\) and corresponding weights \(c_1,\ldots ,c_m>0\) form a John’s decomposition of the identity, then \(\{w_1,\ldots ,w_m\}^*\subset d\mathbf {B}\), cf. [1] or [7, Thm. 5.1]. By polarity, we also obtain that \(\frac{1}{d}\mathbf {B}\subset {\text {conv}}(\{w_1,\ldots ,w_m\})\).

One can verify that if \(\Delta \) is a regular simplex in \({\mathbb {R}}^{d}\) such that the ball \(\mathbf {B}\) is the largest volume ellipsoid in \(\Delta \), then

$$\begin{aligned} {\text {vol}}\left( \Delta \right) =\frac{d^{d/2}(d+1)^{(d+1)/2}}{d!}. \end{aligned}$$
(2)

We will use the following form of the Dvoretzky–Rogers lemma [6].

Lemma 1.4

(Dvoretzky–Rogers lemma) Assume that \(w_1,\ldots ,w_m\in {\text {bd}}{\mathbf {B}}\) and \(c_1,\ldots ,c_m>0\) form a John’s decomposition of the identity. Then there is an orthonormal basis \(z_1,\ldots ,z_d\) of \({\mathbb {R}}^{d}\), and a subset \(\{v_1,\ldots ,v_d\}\) of \(\{w_1,\ldots ,w_m\}\) such that

$$\begin{aligned} v_i\in {\text {span}}\{z_1,\ldots ,z_i\} \;\; \text{ and } \;\; \sqrt{\frac{d-i+1}{d}}\le \left\langle v_i,z_i\right\rangle \le 1 \;\; \text{ for } \text{ all } i=1,\ldots ,d.\;\; \end{aligned}$$
(3)

This lemma is usually stated in the setting of John’s theorem, that is, when the vectors are contact points of a convex body K with its maximal volume ellipsoid, which is \(\mathbf {B}\). And often, it is assumed in the statement that K is symmetric about the origin, see for example [4]. Since we make no such assumption (in fact, we make no reference to K in the statement of Lemma 1.4), we give a proof in Sect. 4.

Fig. 1
figure 1

Finding the ellipsoid \(E_2\)

Full size image

2 Proof of Theorem 1.1

Without loss of generality, we may assume that \(\mathcal {F}\) consists of closed half-spaces, and also that \({\text {vol}}\left( \cap \mathcal {F}\right) <\infty \), that is, \(\cap \mathcal {F}\) is a convex body in \({\mathbb {R}}^{d}\). As shown in [3], by continuity, we may also assume that \(\mathcal {F}\) is a finite family, that is \(P=\cap \mathcal {F}\) is a d-dimensional polyhedron.

The problem is clearly affine invariant, so we may assume that \(\mathbf {B}\subset P\) is the ellipsoid of maximal volume in P.

By Lemma 1.3, there are contact points \(w_1,\ldots ,w_m\in {\text {bd}}{\mathbf {B}}\cap {\text {bd}}{P}\) (and weights \(c_1,\ldots ,c_m>0\)) that form a John’s decomposition of the identity. We denote their convex hull by \(Q={\text {conv}}\{w_1.\ldots ,w_m\}\). Lemma 1.4 yields that there is an orthonormal basis \(z_1,\ldots ,z_d\) of \({\mathbb {R}}^{d}\), and a subset \(\{v_1,\ldots ,v_d\}\) of the contact points \(\{w_1,\ldots ,w_m\}\) such that (3) holds.

Let \(S_1={\text {conv}}\{o,v_1,v_2,\ldots ,v_d\}\) be the simplex spanned by these contact points, and let \(E_1\) be the largest volume ellipsoid contained in \(S_1\). We denote the center of \(E_1\) by u. Let \(\ell \) be the ray emanating from the origin in the direction of the vector \(-u\). Clearly, the origin is in the interior of Q. In fact, by the remark following Lemma 1.3, \(\frac{1}{d}\mathbf {B}\subset Q\). Let w be the point of intersection of the ray \(\ell \) with \({\text {bd}}Q\). Then \(|w|\ge 1/d\). Let \(S_2\) denote the simplex \(S_2={\text {conv}}\{w,v_1,v_2,\ldots ,v_d\}\). See Fig. 1.

We apply a contraction with center w and ratio \(\lambda =\frac{|w|}{|w-u|}\) on \(E_1\) to obtain the ellipsoid \(E_2\). Clearly, \(E_2\) is centered at the origin and is contained in \(S_2\). Furthermore,

$$\begin{aligned} \lambda =\frac{|w|}{|u|+|w|}\ge \frac{|w|}{1+|w|}\ge \frac{1}{d+1}. \end{aligned}$$
(4)

Since w is on \({\text {bd}}Q\), by Caratheodory’s theorem, w is in the convex hull of some set of at most d vertices of Q. By re-indexing the vertices, we may assume that \(w\in {\text {conv}}\{w_1,\ldots ,w_k\}\) with \(k\le d\). Now,

$$\begin{aligned} E_2\subset S_2\subset {\text {conv}}\{w_1,\ldots ,w_k, v_1,\ldots ,v_d\}. \end{aligned}$$
(5)

Let \(X=\{w_1,\ldots ,w_k, v_1,\ldots ,v_d\}\) be the set of these unit vectors, and let \(\mathcal {G}\) denote the family of those half-spaces which support \(\mathbf {B}\) at the points of X. Clearly, \(|\mathcal {G}|\le 2d\). Since the points of X are contact points of P and \(\mathbf {B}\), we have that \(\mathcal {G}\subseteq \mathcal {F}\). By (5),

$$\begin{aligned} \cap \mathcal {G}=X^{*}\subset E_2^{*}. \end{aligned}$$
(6)

By (3),

$$\begin{aligned} {\text {vol}}\left( S_1\right) \ge \frac{1}{d!}\cdot \frac{\sqrt{d!}}{d^{d/2}}=\frac{1}{\sqrt{d!}d^{d/2}}. \end{aligned}$$
(7)

Since \(\mathbf {B}\subset \cap \mathcal {F}\), by (6) and (4), (2), (7) we have

$$\begin{aligned} \frac{{\text {vol}}\left( \cap \mathcal {G}\right) }{{\text {vol}}\left( \cap \mathcal {F}\right) }\le & {} \frac{{\text {vol}}\left( E_2^{*}\right) }{{\text {vol}}\left( \mathbf {B}\right) } =\frac{{\text {vol}}\left( \mathbf {B}\right) }{{\text {vol}}\left( E_2\right) } \le (d+1)^d\frac{{\text {vol}}\left( \mathbf {B}\right) }{{\text {vol}}\left( E_1\right) } = (d+1)^d\frac{{\text {vol}}\left( \Delta \right) }{{\text {vol}}\left( S_1\right) }\nonumber \\= & {} \frac{d^{d/2}(d+1)^{(3d+1)/2}}{d!{\text {vol}}\left( S_1\right) } =\frac{d^dd^{3d/2}e^{3/2}(d+1)^{1/2}}{(d!)^{1/2}} \le e^{d+1}d^{2d+\frac{1}{2}}, \end{aligned}$$
(8)

where \(\Delta \) is as defined above (2). This completes the proof of Theorem 1.1.

Remark 2.1

In the proof, in place of the Dvoretzky–Rogers lemma, we could select the d vectors \(v_1,\ldots ,v_d\) from the contact points randomly: picking \(w_i\) with probability \(c_i/d\) for \(i=1,\ldots ,m\), and repeating this picking independently d times. Pivovarov proved (cf. [9, Lem. 3]) that the expected volume of the random simplex \(S_1\) obtained this way is the same as the right hand side in (7).

3 A Simple Lower Bound for v(d)

We outline a simple proof that one cannot hope a better bound in Theorem 1.1 than \(d^{d/2}\) in place of \(d^{2d+1/2}\). Indeed, consider the Euclidean ball \(\mathbf {B}\), and a family \(\mathcal {F}\) of (very many) supporting closed half space of \(\mathbf {B}\) whose intersection is very close to \(\mathbf {B}\). Suppose that \(\mathcal {G}\) is a subfamily of \(\mathcal {F}\) of 2d members. Denote by \(\sigma \) the Haar probability measure on the sphere \(R\mathbb {S}^{d-1}\), where \(R=(d/(2\ln d))^{\frac{1}{2}}\). Let \(H\in \mathcal {G}\) be one of the half spaces. Then

$$\begin{aligned} \sigma (R\mathbb {S}^{d-1}\setminus H)\le \exp \Bigg ({\frac{-d}{2R^2}}\Bigg ) \le 1/(4d). \end{aligned}$$

It follows that

for any \(\varepsilon >0\) if d is large enough.

4 Proof of Lemma 1.4

We follow the proof in [4].

Claim 4.1

Assume that \(w_1,\ldots ,w_m\in {\text {bd}}{\mathbf {B}}\) and \(c_1,\ldots ,c_m>0\) form a John’s decomposition of the identity. Then for any linear map \(T:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) there is an \(\ell \in \{1,\ldots ,m\}\) such that

$$\begin{aligned} \left\langle w_\ell ,Tw_\ell \right\rangle \ge \frac{{\text {tr}}T}{d}, \end{aligned}$$
(9)

where \({\text {tr}}T\) denotes the trace of T.

For matrices \(A,B\in \mathbb {R}^{d\times d}\) we use \(\left\langle A,B\right\rangle ={\text {tr}}\left( AB^T\right) \) to denote their Frobenius product.

To prove the claim, we observe that

Since \(\sum _{i=1}^m c_i=d\), the right hand side is a weighted average of the values \(\left\langle Tw_i,w_i\right\rangle \). Clearly, some value is at least the average, yielding Claim 4.1.

We define \(z_i\) and \(v_i\) inductively. First, let \(z_1=v_1=w_1\). Assume that, for some \(k<d\), we have found \(z_i\) and \(v_i\) for all \(i=1,\ldots ,k\). Let \(F={\text {span}}\{z_1,\ldots ,z_k\}\), and let T be the orthogonal projection onto the orthogonal complement \(F^{\perp }\) of F. Clearly, \({\text {tr}}T=\dim F^{\perp }=d-k\). By Claim 4.1, for some \(\ell \in \{1,\ldots ,m\}\) we have

$$\begin{aligned} |Tw_\ell |^2=\left\langle Tw_\ell ,w_\ell \right\rangle \ge \frac{d-k}{d}. \end{aligned}$$

Let \(v_{k+1}=w_\ell \) and \(z_{k+1}=\frac{Tw_\ell }{|Tw_\ell |}\). Clearly, \(v_{k+1}\in {\text {span}}\{z_1,\ldots ,z_{k+1}\}\). Moreover,

$$\begin{aligned} \left\langle v_{k+1},z_{k+1}\right\rangle =\frac{\left\langle Tw_\ell ,w_\ell \right\rangle }{|Tw_\ell |}= \frac{|Tw_\ell |^2}{|Tw_\ell |}=|Tw_\ell |\ge \sqrt{\frac{d-k}{d}}, \end{aligned}$$

finishing the proof of Lemma 1.4.

Note that in this proof, we did not use the fact that, in a John’s decomposition of the identity, the vectors are balanced, that is \(\sum _{i=1}^m c_iw_i=o\).