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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|05EGHEQZAW]], let $S \\subseteq V$ be a [[convex set|0K06X9QXDX]], and let $f \\colon S \\to \\R$ be a function.\nThe function $f$ is *quasiconvex* if\n$$f(t x + (1 - t) y) \\le \\max\\set{f(x), f(y)}$$\nfor all $x, y \\in S$ and all $t \\in [0, 1]$.\n\n## Remarks\n\nEquivalently, for each $a \\in \\R$ the sublevel set $\\set{x \\in S : f(x) \\le a}$ is convex; compare [[level sets|0Y5Q8Q9WJ7]].\nThe function $f$ is *quasiconcave* if $-f$ is quasiconvex.\nA [[convex function|0M1GJSWZDA]] is quasiconvex, because $t f(x) + (1 - t) f(y) \\le \\max\\set{f(x), f(y)}$.\n\nExample: $f(x) = \\sqrt{\\abs{x}}$ on $\\R$.\nIts sublevel sets are $\\emptyset$ for $a \u003c 0$ and $[-a^2, a^2]$ for $a \\ge 0$, all convex.\nIt is not convex, since $f(1/2) = \\sqrt{1/2} \u003e 1/2 = \\frac{1}{2} f(0) + \\frac{1}{2} f(1)$.\n\nNon-example: $f(x) = -x^2$ on $\\R$.\nWith $x = -1$, $y = 1$ and $t = 1/2$, one gets $f(0) = 0 \u003e -1 = \\max\\set{f(-1), f(1)}$.\n",
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