In the Euclidean plane $mathbb{R}^{2}$, we define the Minkowski difference$mathcal{K}-mathcal{L}$ of two arbitrary convex bodies $mathcal{K}$,$mathcal{L}$ as a rectifiable closed curve $mathcal{H}_{h}subset mathbb{R}^{2}$ that is determined by the difference $h=h_{mathcal{K}}-h_{mathcal{L}} $ of their support functions. This curve $mathcal{H}_{h}$ iscalled thehedgehog with support function $h$. More generally, the object of hedgehogtheory is to study the Brunn--Minkowski theory in the vector space ofMinkowski differences of arbitrary convex bodies of Euclidean space $mathbb{R}^{n+1}$, defined as (possibly singular and self-intersecting) hypersurfacesof $mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i)studying convex bodies by splitting them into a sum in order to reveal theirstructure; (ii) converting analytical problems intogeometrical ones by considering certain real functions as supportfunctions.The purpose of this paper is to give a detailed study of planehedgehogs, which constitute the basis of the theory. In particular:(i) we study their length measures and solve the extension of theChristoffel--Minkowski problem to plane hedgehogs; (ii) wecharacterize support functions of plane convex bodies among supportfunctions of plane hedgehogs and support functions of plane hedgehogs amongcontinuous functions; (iii) we study the mixed area ofhedgehogs in $mathbb{R}^{2}$ and give an extension of the classical Minkowskiinequality (and thus of the isoperimetric inequality) to hedgehogs.
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