This very interesting paper presents a breakthrough in an old problem motivated by the classical H. Whitney extension papers [Trans. Am. Math. Soc. 36, 63–89 (1934; Zbl 0008.24902 and JFM 60.0217.01), 369–387 (1934; Zbl 0009.20803 and JFM 60.0217.02)]. In rough outline, the problem can be formulated as follows.
Let \(B\) be a Banach space of “smooth” functions on \(\mathbb{R}^n\) and \(X\subset \mathbb{R}^n\).
Problem. Characterize continuous functions on \(X\) admitting extensions to functions of the space \(B\).
“Smoothness” means either the existence of continuous derivatives of fixed order or a prescribed behaviour of difference characteristics (moduli of continuity). Usually one considers “mixed” definitions where the derivatives of higher order have such a behaviour. The typical examples are the classical Sobolev and Besov spaces \(W^k_\infty\) and \(B_\infty^s\). The last two families are parts of a more general family consisting of Lipschitz spaces of higher order. A common member of this family is defined by the seminorm \[ |f|_{\Lambda_\omega^k}:=\sup \left\{\frac{\bigl|\Delta_h^kf(x)\bigr|}{\omega \bigl(|h|\bigr)}\,\biggl|\,x, \lambda\in\mathbb{R}^n\right\}. \] Here \(\Delta_h^k:= (\Delta_h)^k\) is the \(k\)-th difference and \(\omega\) is a homeomorphism of \([0,+\infty)\) such that \(t\mapsto\omega(t^{1/k})\) is a concave function. Note that \(W^k_\infty=\Lambda^k_\omega\) with \(\omega(t)=t^k\) and \(B^\rho_\infty= \Lambda_\omega^k\) with \(\omega(t)=t^\rho\), \(0< \rho<K\).
Finiteness Conjecture (Yu. A. Brudnyi, 1983): Given the space \(\Lambda_\omega^k\) and subset \(X\subset \mathbb{R}^n\), there is an integer \(N=N(k,n)\) with the following property.
A continuous function \(f:X\to\mathbb{R}\) belongs to the trace space \(\Lambda^k_\omega(X)\) iff for every subset \(Y\subset X\) consisting of \(N\) points there is a function \(f_y\in\Lambda_\omega^k\) which agrees with \(f\) on \(Y\) and such that \[ \sup\left\{|f_y|_{\Lambda_\omega^k} \mid Y\subset X,\text{card}\,Y=N \right\}<\infty. \] The optimal \(N\) is denoted by \(N(\Lambda^k_\omega)\).
The Whitney extension method presented in his second paper leads, in particular, to the sharp results for Sobolev and Besov spaces on \(\mathbb{R}\). Namely, \[ N\bigl( W_\infty^k(\mathbb{R})\bigr)=k+1\text{ and }N\bigl(B_\infty^\rho(\mathbb{R}) \bigr)=\widetilde k \] where \(\widetilde k\) is the least integer bigger than \(\rho\).
The similar result for the space \(\Lambda_\omega^k(\mathbb{R})\) (with the optimal constant \(k+1)\) follows from the theorem of I. A. Shevchuk [Anal. Math. 10, 249–273 (1984; Zbl 0596.41049)], see also his book “Approximation by polynomials and traces of functions continuous on a segment of the real line” (Russian) [Kiev, Naukova Dumka (1992)].
The multidimensional case is much more complicated. The reviewer proposed the following program to prove the conjecture. Using a local polynomial description of the space \(\Lambda_\omega^k(\mathbb{R}^n)\) [Yu. Brudnyi, Math. USSR, 11, 157–170 (1970); translation from Mat. Sb., N. Ser. 82(124), 175–191 (1970; Zbl 0204.13501)] and the extension method preserving local polynomial approximation [Yu. A. Brudnyi, Funct. Anal. Appl. 4, 252–253 (1970); translation from Funkts. Anal. Prilozh. 4, No. 3, 97–98 (1970; Zbl 0221.46008)], it is possible to characterize the trace space \(\Lambda_\omega^k(X)\) in terms of local polynomial approximation [Yu. A. Brudnyi and P. Shvartsman, Issled. Teor. Funkts. Mnogikh Veshchestv. Perem. 1982, 16–24 (1982; Zbl 0598.41033)]. The idea is to reformulate this characteristic in geometric terms which, in particular, leads to the following conjecture yet unproven.
Conjecture (Yu. A. Brudnyi, 1984): Let \(\varphi\) be a map from a metric space \(({\mathcal M},d)\) to compact subsets of \(\mathbb{R}^n\). Then \(\varphi\) admits a Lipschitz selection \(f:{\mathcal M}\to\mathbb{R}^n\) iff the restriction of \(\varphi\) to every \(2^n\) point subset \(S\subset{\mathcal M}\) has a Lipschitz selection \(f_s\) and the Lipschitz constants \(\text{Lip}(f_s)\) are uniformly bounded:
The first considerable success was achieved in the PhD thesis of P. Shvartsman. He proved a special case of the conjecture for \(\varphi\) taking values in the set of all affine subspaces of \(\mathbb{R}^n\), and derived based on that the following beautiful result
\[ N\left( \Lambda^2_\omega(\mathbb{R}^n)\right)=3.2^{n-1}. \]
The abridged version of P. A. Shvartsman’s proof appeared in [Sib. Math. J. 28, No. 5, 853–863 (1987); translation from Sib. Mat. Zh. 28, No. 5(165), 203–215 (1987; Zbl 0634.46025)]. The method is also applied to the space \(C^{1,\omega} (\mathbb{R}^n)\) where the finiteness constant is the same [Yu. Brudnyi and P. Shvartsman, Trans. Am. Math. Soc. 353, No. 6, 2487–2512 (2001; Zbl 0973.46025)].
The further advance towards higher than second smoothness, e.g., \(W_\infty^k (\mathbb{R}^k)\) with \(k>2\) or \(B^s_\infty(\mathbb{R}^k)\) with \(\rho\geq 2\) requires new ideas. The first result in this direction is Ch. Fefferman’s Theorem A of the paper under review; it asserts that the Finiteness Conjecture is valid for the Sobolev space \(W^k_\infty(\mathbb{R}^n)\). The proof is a very impressive coup de force. It skillfully combines methods of modern real analysis, algebraic geometry (resolution of singularities) and convex geometry. One hopes that the clarification of the basic ideas of this proof may lead to the discovery of new important results in the now actively developing (but unnamed) field devoted to problems with incomplete information. Problems of this kind have appeared in a wide range of research areas (mathematical physics, numerical analysis, signal and image processing, the theory of computation, learning theory, etc.).