Let $x_1,\dots,x_n\in[-1,1]$ be distinct nodes and let\[l_k(x)=\prod_{i\neq k}\frac{x-x_i}{x_k-x_i}\]denote the associated Lagrange interpolation polynomials. Erd\H{o}s posed the problem of minimizing the functional\[I(x_1,\dots,x_n)=\int_{-1}^1 \sum_{k=1}^n |l_k(x)|^2\,dx\]and determining its asymptotic behavior as $n\to\infty$. It was known that\[2-O\!\left(\frac{(\log n)^2}{n}\right)\le \inf I \le 2-\frac{2}{2n-1},\]with the upper bound attained by nodes related to Legendre polynomials. In this paper, we place Erd\H{o}s’s problem within the classical framework of minimal-norm interpolation. We interpret $I$ as the squared Hilbert--Schmidt norm of the associated Lagrange interpolation operator and recall that asymptotic minimizers are constrained by the structure and uniqueness theory of minimal $L^2$-norm interpolation schemes. Building on this foundation, we develop a variational approach based on Christoffel functions, orthogonal polynomial asymptotics, and entropy methods to resolve the problem asymptotically. Our main contributions are as follows:\begin{enumerate}\item[(i)] We prove that any asymptotically minimizing sequence of nodes must equidistribute with respect to the arcsine measure on ([-1,1]). ```\item[(ii)] We establish a sharp \(O(1/n)\) lower bound, improving the longstanding \(O((\log n)^2/n)\) estimate of Erd\H{o}s–Szabados–Varma–V\'ertesi. \item[(iii)] We show that asymptotic minimizers are rigid and structurally constrained, in accordance with classical minimal-norm interpolation theory. \item[(iv)] We identify that the leading correction arises from microscopic endpoint regions and formulate an \emph{entropy rigidity hypothesis} connecting deterministic minimization to equilibrium log-gas behavior. \item[(v)] Under a conjectured \emph{endpoint universality} principle for discrete Christoffel functions, we derive the first-order asymptotic expansion\[\inf I = 2 - \frac{c}{n} + o\!\left(\frac{1}{n}\right),\]with an explicit constant \(c>0\) expressed in terms of the Airy kernel. \item[(vi)] We show that the Legendre–integral nodes are asymptotically optimal and support all theoretical predictions with detailed numerical experiments, including verification of edge rigidity and Airy-type endpoint scaling.