Problem 1129

For $x_1,\ldots,x_n\in [-1,1]$ let

l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},

which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$ .

Describe which choice of $x_i$ minimise

\Lambda(x_1,\ldots,x_n)=\max_{x\in [-1,1]} \sum_k \lvert l_k(x)\rvert.

Source: erdosproblems.com/1129

Analysis
Polynomials

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

The site labels the problem PROVED (page last edited 23 January 2026, label accessed 2026-09-04), and the community database at teorth/erdosproblems lists its status as proved (Lean) as of its last update on 2026-09-16, pointing to a third-party Lean development described below. The site credits the characterization of the minimizing nodes to de Boor and Pinkus [dBPi78], after Kilgore and Cheney [KiCh76] and Kilgore [Ki77], and the four-node optimum to Rack and Vajda [RaVa15]; the accepted claim pages are de Boor and Pinkus 1978, which records the paper's convention that the endpoints are nodes, Kilgore 1978, an independent proof of Bernstein's conjecture in the same issue, and the partial Rack and Vajda 2015, which describes every four-node minimizer. The question asks for a description, so the derived claim value is answered.

Be31: S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degré n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.
Br80: Brutman, L., On the polynomial and rational projections in the complex plane. SIAM J. Numer. Anal. (1980), 366-372.
BrPi80: Brutman, L. and Pinkus, A., On the Erdős conjecture concerning minimal norm interpolation on the unit circle. SIAM J. Numer. Anal. (1980), 373-375.
Er47: Erdős, P., Some remarks on polynomials. Bull. Amer. Math. Soc. 53 (1947), 1169-1176.
Er61c: Erdős, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.
Er67: Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) 10 (33) (1968), 65-73.
Fa14: G. Faber, Über die interpolatorische Darstellung stetiger Funktionen. Jahresb. der Deutschen Math. Ver. (1914), 190-210.
Ki77: Kilgore, T. A., Optimization of the norm of the Lagrange interpolation operator. Bull. Amer. Math. Soc. (1977), 1069-1071.
KiCh76: Kilgore, T. A. and Cheney, E. W., A theorem on interpolation in Haar subspaces. Aequationes Math. (1976), 391-400.
RaVa15: Rack, Heinz-Joachim and Vajda, Robert, Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant. Stud. Univ. Babeş-Bolyai Math. 60 (2015), no. 2, 151-171.
dBPi78: de Boor, Carl and Pinkus, Allan, Proof of the conjectures of Bernstein and Erdős concerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289-303.

Problem 1129

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