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  "body": "\n## Counterexample\n\nFor a [[convex polytope|300K6CB1KH]] $P$, the graph of $P$ has the vertices of $P$ as vertices and the edges of $P$ as edges.\nThe diameter of $P$ is the largest [[distance|0QV9VNNB4Z]] in this graph between two vertices.\nThe Hirsch conjecture asserted: if $d \\ge 2$ and $P$ is a $d$-dimensional polytope with $n$ [[facets|DPDXN2FJC4]], then the diameter of $P$ is at most $n - d$.\nThe conjecture is false: in 2010 Francisco Santos constructed a $43$-dimensional polytope with $86$ facets whose diameter is at least $44$, while $n - d = 43$.\n\n## Remarks\n\nSantos's construction first builds a spindle: a polytope with two special vertices whose distance exceeds the dimension.\nA known reduction, which adds facets and dimensions in step, then turns it into a counterexample.\nFor unbounded polyhedra the analogous statement had already been disproved by Victor Klee and David Walkup in 1967.\nThe bound $n - d$ is attained by the $d$-cubes, which have $n = 2d$ facets and diameter $d$.\n\nThe question matters for [[linear programming|VNMXB0J2KA]], since the simplex method walks along edges of the feasible polytope.\nSo the diameter bounds from below the worst number of steps of any edge-following pivot rule.\nWhether the diameter is bounded by a polynomial in $n$ and $d$ remains open.\nThe best general bound known is quasi-polynomial, of the form $n^{\\log_2 d + 2}$, due to Gil Kalai and Daniel Kleitman.\n\n## Explanations\n\n### L1 human\n\nWalk from corner to corner along the edges of a polytope.\nThe Hirsch conjecture claimed that any two corners are joined by a short walk: at most the number of walls minus the dimension.\nFor half a century nobody could prove or refute it.\nSantos found a polytope in $43$ dimensions where some walk must be longer, so the conjecture is false, though only by a little.\n",
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    "name": "Hirsch conjecture",
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  "msc": [
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