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  "body": "\n## Summary\n\nThe Minkowski functional, or gauge, of an [[absorbing set|NM1BCK3ZX9]] $K$ in a vector space is a function on the space.\nIts value at $x$ is the smallest factor by which $K$ must be scaled to contain $x$.\nFor convex sets it is sublinear, and for convex [[balanced|2TDSHJJYS4]] sets it is a [[seminorm|FP5GVCN8GV]].\nIt is the bridge between convex neighbourhoods of $0$ and seminorms, and so it is the main tool for describing [[locally convex spaces|B6GY2KJMWW]] by seminorms.\n\n## Explanations\n\n### L1 human\n\nTake a convex shape around the origin, for example a disc or a square.\nTo measure a point $x$, blow up or shrink the shape until its boundary just reaches $x$; the scaling factor is the gauge of $x$.\nIf the shape is the unit disc, the gauge is the ordinary length.\nIf the shape is the square $[-1, 1]^2$, the gauge of $(x, y)$ is $\\max(\\abs{x}, \\abs{y})$.\nSo every symmetric convex shape gives its own way of measuring size.\n",
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