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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[complete metric space|0BY52JW6NK]].\nLet $f \\colon X \\to (-\\infty, +\\infty]$ be [[lower semicontinuous|RN4G8SGP6P]], bounded below, and not identically $+\\infty$.\nLet $\\varepsilon \u003e 0$ and $\\lambda \u003e 0$, and let $x_0 \\in X$ satisfy $f(x_0) \\le \\inf_X f + \\varepsilon$.\nThen there is a point $v \\in X$ such that:\n\n1. $f(v) \\le f(x_0)$;\n2. $d(x_0, v) \\le \\lambda$;\n3. $f(w) \u003e f(v) - \\dfrac{\\varepsilon}{\\lambda} d(v, w)$ for every $w \\in X$ with $w \\ne v$.\n\n## Remarks\n\nThe proof builds a sequence $x_0, x_1, x_2, \\ldots$.\nEach $x_{n+1}$ is chosen in the set of points $y$ with $f(y) \\le f(x_n) - (\\varepsilon / \\lambda) d(x_n, y)$, with $f(x_{n+1})$ close to the infimum of $f$ on that set.\nThese sets are closed and nested, their diameters tend to $0$, and completeness gives a limit point $v$, which has the three properties.\nTaking $\\lambda = 1$ on a Banach space with $f$ Gateaux differentiable gives a point $v$ with $f(v) \\le \\inf f + \\varepsilon$ and $\\norm{f'(v)} \\le \\varepsilon$: an almost minimizer with small derivative.\nWith this, a $C^1$ function that is bounded below and satisfies the [[Palais–Smale condition|4724KRNHAT]] attains its minimum.\nCompleteness is needed: on $X = (0, 1]$ with $f(x) = x$, $\\varepsilon = 1/2$ and $\\lambda = 1$, every $v$ violates clause 3 for any $w \u003c v$, because $f(w) = v - (v - w) \u003c v - \\tfrac12 (v - w)$.\n\n## Explanations\n\n### L1 human\n\nA function that is bounded below need not have a minimum point.\nEkeland's principle says that one can always find a point that is almost as good as the infimum and that is a strict minimum of a slightly tilted function.\nThe tilt is a cone of slope $\\varepsilon / \\lambda$ around the point.\nSo near-minimizers can always be taken to be \"stable\" in a precise sense.\n\n### L2 human\n\nThe principle is a basic tool of nonsmooth analysis and of critical point theory.\nIt produces Palais–Smale sequences for functionals bounded below, and it gives short proofs of the Caristi fixed point theorem and of the Bishop–Phelps theorem.\nConversely, a metric space in which the conclusion holds for all such functions is complete, so the principle characterizes completeness.\n",
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