1 Introduction

In this paper we address the following problem (cf., for example, [2, Problem 3.2, p. 125]).

Problem 1

Let \(2\le k\le d-1\). Assume that K and L are convex bodies in \({{\mathbb {R}}}^d\) such that the projections K|H and L|H are congruent for all \(H\in {{\mathcal {G}}}(d,k)\). Is K a translate of \(\pm L\)?

Here we say that K|H, the projection of K onto H, is congruent to L|H if there exists an orthogonal transformation \(\varphi \in O(k,H)\) in H such that \(\varphi (K|H)\) is a translate of L|H; \({{\mathcal {G}}}(d,k)\) stands for the Grassmann manifold of all k-dimensional subspaces in \({{\mathbb {R}}^d}\).

Recently, Myroshnychenko [6] together with the author gave an affirmative answer to Problem 1 in the class of polytopes. We refer the reader to [1, 3, 5, 7] and [8], for the history and some partial results related to Problem 1.

Our first result is

Theorem 1

Let K and L be two convex bodies in \({{\mathbb {R}}^5}\). Assume that for every \(\xi \in S^4\) the projections \(K|\xi ^{\perp }\) and \(L|\xi ^{\perp }\) are directly SU(2)-congruent, i.e., for every \(\xi \in S^4\) there is a rotation \(\varphi _{\xi }\in SU(2,\xi ^{\perp })\) for some complex structure in \(\xi ^{\perp }\) and a vector \(a_{\xi }\in \xi ^{\perp }\) such that

$$\begin{aligned} \varphi _{\xi }(K|\xi ^{\perp })+a_{\xi }=L|\xi ^{\perp }. \end{aligned}$$
(1)

Then \(K+b=L\) or \(-K+b=L\) for some \(b\in {{\mathbb {R}}^5}\), provided that the set of diameters of one of the bodies is contained in a finite union of two-dimensional subspaces of \({{\mathbb {R}}^5}\).

We obtain Theorem 1 as a consequence of a more general statement about a functional equation on the unit sphere. Let

$$\begin{aligned} M(g_e)=\left\{ x\in S^4:\,g_e(x)=\max \limits _{S^4}g_e\right\} \end{aligned}$$
(2)

be the set of directions of the maxima of the even part of a continuous function g defined on \(S^4\). We have

Theorem 2

Let f and g be two continuous functions on \(S^{4}\). Assume that for every \(\xi \in S^4\) there is a rotation \(\varphi _{\xi }\in SU(2,\xi ^{\perp })\) for some complex structure in \(\xi ^{\perp }\) and a vector \(a_{\xi }\in \xi ^{\perp }\) such that

$$\begin{aligned} f(\varphi _{\xi }(x))+a_{\xi }\cdot x=g(x)\qquad \forall x\in S^4\cap \xi ^{\perp }. \end{aligned}$$
(3)

Then there exists \(b\in {{\mathbb {R}}^5}\) such that \(f(x)+b\cdot x=g(x)\) for all \(x\in S^{4}\) or \(f(-x)+b\cdot x=g(x)\) for all \(x\in S^{4}\), provided that \(M(g_e)\) is contained in a finite union of large 1-dimensional circles of \(S^4\).

The paper is organized as follows. In the next section we recall some definitions and prove several auxiliary Lemmata that will be used later. We prove Theorems 2 and 1 in Sects. 3 and 4.

1.1 Notation

We denote by \(S^{4}=\{x\in {\mathbb {R}}^5:\,|x|=1\}\) the set of all unit vectors in the Euclidean space \({\mathbb {R}}^5\). For any unit vector \(\xi \in S^{4}\) we let \(\xi ^{\perp }\) to be the orthogonal complement of \(\xi \) in \({\mathbb {R}}^{5}\), i.e., the set of all \(x \in {\mathbb {R}}^{5}\) such that \(x \cdot \xi = 0\); here \(x \cdot \xi \) stands for a usual scalar product of x and \(\xi \) in \({\mathbb {R}}^5\). The notation for the orthogonal group O(k) and the special orthogonal group SO(k), \(k\ge 2\), is standard; \(span(a_1, a_2,\ldots , a_m)\) stands for a m-dimensional subspace that is a linear span of linearly independent vectors \(a_1,\ldots , a_m\), \(m\ge 1\). We will write \(f_e\) and \(f_o\) for the even and odd parts of the function f,

$$\begin{aligned} f_e(x)=\frac{f(x)+f(-x)}{2},\qquad f_o(x)=\frac{f(x)-f(-x)}{2},\quad x\in {{\mathbb {R}}^5}. \end{aligned}$$

2 Auxiliary definitions and results

We introduce a complex structure in \({{\mathbb {R}}^4}\) by identifying it with \({{\mathbb {C}}^2}\). We will say that two bodies A and B in \({{\mathbb {R}}^4}={{\mathbb {C}}^2}\) are directly SU(2)-congruent if there exists a vector \(a\in {{\mathbb {R}}^4}\) and a SU(2)-rotation \(\varphi _{{{\mathbb {R}}^4}}\) such that \(\varphi (A)+a=B\).

Consider any 4-dimensional subspace \(\xi ^{\perp }\) of \({{\mathbb {R}}^5}\) orthogonal to \(\xi \in S^4\). We say that \(\varphi _{\xi }\in SO(4, \xi ^{\perp })\), meaning that there exists a choice of an orthonormal basis in \({{\mathbb {R}}}^5\) and a rotation \(\Phi \in SO(5)\), with a matrix written in this basis, such that the action of \(\Phi \) on \(\xi ^{\perp }\) is the rotation \(\varphi _{\xi }\) in \(\xi ^{\perp }\), and the action of \(\Phi \) on \(l(\xi )=(\xi ^{\perp })^{\perp }\) is trivial, i.e., \(\Phi (y)=y\) for every \(y\in l(\xi )\).

We say that a rotation \(\varphi _{\xi }\) is in \(SU(2,\xi ^{\perp })\) if its matrix \(A_{\xi }\) with respect to a certain basis in \(\xi ^{\perp }``=''{{\mathbb {R}}^4}``=''{{\mathbb {C}}}^2\) is of the form (see [9], page 130):

$$\begin{aligned} A_{\xi }=\begin{bmatrix} e^{i\varphi }&\quad 0 \\ 0&\quad e^{-i\varphi } \end{bmatrix},\qquad \varphi \in [-\pi ,\pi ]. \end{aligned}$$

Here the invariant subspaces of \(\varphi _{\xi }\) (for \(\varphi \ne 0,\pi \)) are the orthogonal complex lines (two-dimensional real subspaces of \(\xi ^{\perp }\)) \(l_1=l_1(\xi )\) and \(l_2=l_2(\xi )\); the restriction \(\varphi _{\xi }|_{l_1}\) is equivalent to a multiplication by \(e^{i\varphi }\), and the restriction \(\varphi _{\xi }|_{l_2}\) is equivalent to a multiplication by \(e^{-i\varphi }\).

We identify \(SU(2,\xi ^{\perp })\) with a subgroup of \(SO(4,\xi ^{\perp })\) of the so-called isoclinic rotations, [11].

Lemma 1

If f and g verify the conditions of Theorem 2, then \(f_e = g_e\) on \(S^4\).

Proof

Comparing the even parts of Eq. (3) we have

$$\begin{aligned} f_e(\varphi _{\xi } (u)) = g_e(u) \quad \text {for any} \quad \xi \in S^4 \quad \text {and any } \quad u \in S^4 \cap \xi ^{\perp }. \end{aligned}$$

Integrating over \(S^4\cap \xi ^{\perp }\) and using the invariance of the Lebesgue measure under rotations, we obtain

$$\begin{aligned} \int \limits _{S^4 \cap \xi ^{\perp }} f_e(\varphi _{\xi } (u)) d\sigma (u) = \int \limits _{S^4 \cap \xi ^{\perp }} f_e(u) d\sigma (u)= \int \limits _{S^4 \cap \xi ^{\perp }} g_e(u) d\sigma (u). \end{aligned}$$

In other words, \(Ff_e=Fg_e\) on \(S^4\), where

$$\begin{aligned} Ff_e(\xi )=\int \limits _{S^4 \cap \xi ^{\perp }} f_e(u) d\sigma (u),\qquad \xi \in S^4, \end{aligned}$$

is the Funk transform on \(S^4\). Since it is injective on even functions (see [4], Corollary 2.7, p. 128), we obtain the desired result. \(\square \)

From now on in this section we will assume that the functions are odd.

Lemma 2

(cf. Lemma 1 [7]). Let \(z\in S^4\) and let \(S^4\cap z^{\perp }=\Lambda _0\cup \Lambda _{\pi }\), where

$$\begin{aligned} \Lambda _0= & {} \left\{ \xi \in S^4\cap z^{\perp }:\, f_o(x)=g_o(x)\qquad \forall x\in S^4\cap \xi ^{\perp }\right\} ,\\ \Lambda _{\pi }= & {} \left\{ \xi \in S^4\cap z^{\perp }:\, -f_o(x)=g_o(x)\qquad \forall x\in S^4\cap \xi ^{\perp }\right\} . \end{aligned}$$

Then \(f_o=g_o\) on \(S^4\) or \(f_o=-g_o\) on \(S^4\).

Proof

Observe that

$$\begin{aligned} \forall x\in S^4, \qquad S^4=\bigcup \limits _{\{\xi \in S^4\cap z^{\perp }\cap x^{\perp }\}}(S^4\cap \xi ^{\perp }). \end{aligned}$$
(4)

Indeed, for any \(y\in S^4\) we take \(\xi \in S^4\cap z^{\perp }\cap x^{\perp }\cap y^{\perp }\) to obtain that \(y\in S^4\cap \xi ^{\perp }\).

Assume that there exists \(x\in S^4\) such that \((S^4\cap z^{\perp }\cap x^{\perp })\subset \Lambda _0\), then, using (4), we see that \(f_o=g_o\) on \(S^4\). Similarly, if there exists \(x\in S^4\) such that \((S^4\cap z^{\perp }\cap x^{\perp })\subset \Lambda _{\pi }\), then, \(f_o=-g_o\) on \(S^4\).

On the other hand, if for any \(x\in S^4\) there exists two directions \(\xi _1\) and \(\xi _2\in S^4\cap z^{\perp }\cap x^{\perp }\), \(\xi _1\ne \pm \xi _2\), such that \(\xi _1\in \Lambda _0\) and \(\xi _2\in \Lambda _{\pi }\), then \(f_o(x)=g_o(x)=-f_o(x)=0\). Hence, \(f_o=g_o=0\) on \(S^4\). \(\square \)

Let \(z\in S^4\). Define

$$\begin{aligned} \Xi _0=\left\{ \xi \in S^4\cap z^{\perp }:\, f_o(x)+a_{\xi }\cdot x=g_o(x)\qquad \forall x\in S^4\cap \xi ^{\perp }\right\} , \end{aligned}$$

and

$$\begin{aligned} \Xi _{\pi }=\left\{ \xi \in S^4\cap z^{\perp }:\, -f_o(x)+a_{\xi }\cdot x=g_o(x)\qquad \forall x\in S^4\cap \xi ^{\perp }\right\} . \end{aligned}$$

Theorem 3

(cf. Theorem 1.3 [5]). Let f and g be two odd continuous functions on \(S^{4}\) and let \(z\in S^4\). Assume that \(S^4\cap z^{\perp }=\Xi _0\cup \Xi _{\pi }\). Then there exists \(b \in {{\mathbb {R}}^5}\) such that for all \(u \in S^{4}\) we have \(g_o(u) = f_o(u) + b \cdot u\), or for all \(u \in S^{4}\) we have \(g_o(u) = -f_o(u) + b \cdot u\).

Proof

Since the proof is very similar to the one of Theorem 1.3, [5], we sketch it briefly. Take \(n=5\) in Theorem 1.3 and Lemma 4.3 [5]. Repeating the argument, we obtain \(S^4\cap z^{\perp }=\Lambda _0\cup \Lambda _{\pi }\) (except an obvious difference with the definitions of \(\Xi _0\) and \(\Xi _{\pi }\) in this note and in [5], Lemmata 3.7 and 3.8 follow without any changes). It remains to apply the previous lemma with the sets \(\Lambda _0\) and \( \Lambda _{\pi }\) that are defined analogously to those in Lemma 4.2, [5], and with the functions \({\tilde{f}}_o\) and \({\tilde{g}}_o\) that appear in the proof of Lemma 4.3 [5]. \(\square \)

3 Proof of Theorem 2

Assume at first that the set of maxima of \(g_e\) consists of two opposite points, i.e.,

$$\begin{aligned} M(g_e)=\left\{ x\in S^4:\,g_e(x)=\max \limits _{S^4}g_e\right\} =\{\pm z\} \end{aligned}$$
(5)

for some \(z\in S^4\). Consider any \(\xi \in S^4\cap z^{\perp }\). We claim that

$$\begin{aligned} M_{\xi }(f_e)=\left\{ x\in S^4\cap \xi ^{\perp }:\,f_e(x)=M(f_e)\right\} =\{\pm z\}. \end{aligned}$$
(6)

To show (6), observe at first that

$$\begin{aligned} \max \limits _{S^4\cap \xi ^{\perp }}f_e=g_e(z). \end{aligned}$$
(7)

Indeed, let \(y\in S^4\cap \xi ^{\perp }\) be such that \(f_e(y)=\max \nolimits _{S^4\cap \xi ^{\perp }}f_e>g_e(z)\). Since the identity

$$\begin{aligned} f_e(\varphi _{\xi }(x))=g_e(x)\qquad \forall x\in S^4\cap \xi ^{\perp }, \end{aligned}$$
(8)

obtained by taking even parts of (3), is equivalent to

$$\begin{aligned} f_e(y)=g_e(\varphi _{\xi }^{-1}(y))\qquad \forall y\in S^4\cap \xi ^{\perp }, \end{aligned}$$
(9)

we see that (9) does not hold, for, \(f_e(y)>g_e(z)\ge g_e(\varphi ^{-1}_{\xi }(y))\). Hence, \(\max \nolimits _{S^4\cap \xi ^{\perp }}f_e\le g_e(z)\). Since \(f_e(\varphi ^{-1}_{\xi }(z))=g_e(z)\), a similar argument shows that \(\max \nolimits _{S^4\cap \xi ^{\perp }}f_e\) may not be smaller than \(g_e(z)\). We have proved (7).

Next, we observe that for each \(\xi \in S^3\cap z^{\perp }\), the set \(M_{\xi }(f_e)\) consists of two opposite points on \(S^4\). Indeed, if the maximum were reached at two points \(y_1\), \(y_2\in S^4\cap \xi ^{\perp }\), \(y_1\ne \pm y_2\), then, using (9), we see that \(g_e\) would reach the maximum at two different points \(\varphi _{\xi }^{-1}(y_1)\) and \(\varphi _{\xi }^{-1}(y_2)\ne \pm \varphi _{\xi }^{-1}(y_1)\). This contradicts (5).

Now we show (6). If it is \(\{\pm y\}\) for some \(y\ne z\), \(y\in S^4\cap \xi ^{\perp }\), we take \(\zeta \in (S^3\cap z^{\perp })\setminus (S^3\cap y^{\perp })\). Since \(y\notin S^4\cap \zeta ^{\perp }\), Eq. (8) may not hold with \(\xi =\zeta \). Thus, (6) holds, and we obtain \(M(f_e)=M(g_e)=\{\pm z\}\).

Using the previous identity and (8), we see that \(\varphi _{\xi }(z)=\pm z\) for all \(\xi \in S^4\cap z^{\perp }\). For, \(\varphi _{\xi }(z)\) must be a point where the maximum of \(f_e\) is reached. Hence, we can assume that for every \(\xi \in S^3\cap z^{\perp }\) the angle of rotation of \(\varphi _{\xi }\in SU(2,\xi ^{\perp })\) is zero or \(\pi \) (since the rotations \(\varphi _{\xi }\) are all isoclinic [11], any ray r in \(\xi ^{\perp }\) emanating from the origin is not parallel to \(\varphi _{\xi }(r)\), unless the angle of rotation is zero or \(\pi \)).

Thus, we can assume that for all \(\xi \in S^4\cap z^{\perp }\), there exists \(a_{\xi }\in \xi ^{\perp }\) such that

$$\begin{aligned} f(x)+a_{\xi }\cdot x=g(x)\qquad \forall x\in S^4\cap \xi ^{\perp }, \end{aligned}$$
(10)

or

$$\begin{aligned} f(-x)+a_{\xi }\cdot x=g(x)\qquad \forall x\in S^4\cap \xi ^{\perp }. \end{aligned}$$
(11)

The proof of Theorem 2 in the case when \(M(g_e)\) consists of a pair of opposite points on \(S^4\) now follows from Lemma 1 and Theorem 3.

Consider the general case. Assume that \(M(g_e)\) is a subset A of finitely many one-dimensional large circles of \(S^4\), \(A\subset \bigcup \nolimits _{j=1}^k{{\mathbb {S}}}_j\), \({{\mathbb {S}}}_j=S^4\cap \Pi _j\), where \(\Pi _j\) is a two-dimensional subspace of \({{\mathbb {R}}}^5\).

Let \(z\in A\) and let \(\xi \in S^4\cap z^{\perp }\). Then, \(\xi ^{\perp }\supset \Pi _j\) if and only if \(\xi \in \Pi _j^{\perp }\), \(j=1, \ldots , k\). Consider

$$\begin{aligned} G_z=(S^4\cap z^{\perp })\setminus \left( \bigcup \limits _{j=1}^k\left( S^4\cap \Pi _j^{\perp }\right) \right. . \end{aligned}$$

For every \(\xi \in G_z\), the subspace \(\xi ^{\perp }\) does not contain any \(\Pi _j\), and we have \(\xi ^{\perp }\cap A=\{\pm z\}\). Then, for any \(\xi \in G_z\), \(M_{\xi }(g_e)=\{\pm z\}\). Repeating the argument of the first part of the proof, we obtain (10), (11) for any \(\xi \in G_z\). Since \(G_z\) is dense in \(S^4\cap z^{\perp }\), we have (10) and (11) for any \(\xi \in S^4\cap z^{\perp }\) (for any \(\xi \in S^4\cap z^{\perp }\) it is enough to consider a sequence of subspaces \(\{\xi ^{\perp }_k\}_{k=1}^{\infty }\), \(\xi _k\in G_z\), \(\xi _k\rightarrow \xi \) as \(k\rightarrow \infty \), for which (10) or (11) holds in the corresponding \(\xi ^{\perp }_k\), and pass to the limit as \(k\rightarrow \infty \); one can use a converging subsequence of \(\{a_{\xi _k}\}_{k=1}^{\infty }\) if necessary). It remains to apply Lemma 1 and Theorem 3.

The proof of Theorem 2 is complete.

4 Proof of Theorem 1

We denote by \(h_K(x)\) the support function of a convex body \(K\subset {{\mathbb {R}}^n}\). For \(x\in {{\mathbb {R}}^n}\) it is defined as \(h_K(x)=\sup \nolimits _{y\in K}x\cdot y\), ([10], page 37), and it is a homogeneous function of degree 1. The width of a set \(A\subset {{\mathbb {R}}^n}\) in the direction \(x\in {{\mathbb {R}}^n}\), is defined as \(\omega _A(x)=h_A(x)+h_A(-x)\). A segment \([z,y]\subset K\) is called a diameter of the convex body K if \(|z-y|=\max \nolimits _{\{\theta \in S^{n-1}\}}\omega _K(\theta )\). We also define \(M(\omega _L|_{S^4})\) as in (2).

We will use the following well-known properties of the support function. For every convex body K,

$$\begin{aligned} h_{K|\xi ^{\perp }}(x)=h_{K}(x) \; \text{ and } \; h_{\varphi _{\xi }(K|\xi ^{\perp })}(x)=h_{K|\xi ^{\perp }}\left( \varphi _{\xi }^{-1}(x)\right) , \quad \forall x \in \xi ^{\perp }, \end{aligned}$$
(12)

(see, for example [2, (0.21), (0.26), pages 17–18]); here \(\varphi _{\xi }^{-1}\) stands for the inverse of \(\varphi _{\xi }\in SO(4,\xi ^{\perp })\).

Theorem 1 can be reformulated in terms of support functions as follows.

Theorem 4

Let K and L be two convex bodies in \({{\mathbb {R}}^5}\). Assume that for every \(\xi \in S^{4}\) there is a rotation \(\varphi _{\xi }\in SU(2,\xi ^{\perp })\) for some complex structure in \(\xi ^{\perp }\) and a vector \(a_{\xi }\in \xi ^{\perp }\) such that

$$\begin{aligned} h_{K|\xi ^{\perp }}\left( \varphi _{\xi }^{-1}(x)\right) +a_{\xi }\cdot x=h_{L|\xi ^{\perp }}(x)\qquad \forall x\in \xi ^{\perp }. \end{aligned}$$
(13)

Assume also that \(M(\omega _L|_{S^4})\) is contained in finitely many 1-dimensional great circles of \(S^4\). Then there exists \(b\in {{\mathbb {R}}^5}\) such that \(h_K(x)+b\cdot x=h_L(x)\) for all \(x\in {{\mathbb {R}}^5}\), or \(h_K(x)+b\cdot x=h_L(-x)\) for all \(x\in {{\mathbb {R}}^5}\).

The proof of Theorems 4 and 1 now follows directly from Theorem 3, provided we take \(f=h_K\) and \(g=h_L\).