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  "body": "\n## Summary\n\nThe Fenchel conjugate, or convex conjugate, of a function $f$ on a vector space is the function $f^*$ on the dual space given by $f^*(y) = \\sup_x \\big(\\ip{y}{x} - f(x)\\big)$.\nIt is always [[convex|07Q7CNM5WF]] and lower semicontinuous, and for proper convex lower semicontinuous $f$ one has $f^{**} = f$.\nIt extends the classical [[Legendre transformation|ZQYMWVT1TT]] to functions that need not be differentiable.\nIt is the basic tool of duality in convex optimization.\n\n## Explanations\n\n### L1 human\n\nThink of the graph of $f$ and of all lines of slope $y$ that stay below it.\nThe highest such line crosses the vertical axis at height $-f^*(y)$.\nSo $f^*$ records, slope by slope, how the graph of $f$ is supported from below.\nFor example, the conjugate of $f(x) = \\frac12 x^2$ is $f^*(y) = \\frac12 y^2$.\nA convex function can be rebuilt from its conjugate, which makes this a kind of \"Fourier transform\" of convex analysis.\n",
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