We study special convex bodies, hedgehogs, and multihedgehogs, extending classical notions such as width. As an application, we construct a simple noncircular algebraic curve of constant width, answering a problem raised by S. Rabinowitz. We explore links between linear projective hedgehogs and Zindler curves, introduce Zindler-type surfaces in \(\mathbb {R}^{4}\) via symplectic area, and motivate a Brunn-Minkowski theory for minimal surfaces. Applying hedgehog theory to analysis, we show the cosine transform is a bounded linear operator on \(\mathbb {S}^{n}\) , implying certain zonoid boundaries are \(\mathrm {C}^2\) -hedgehogs. We define projection hedgehogs, interpret their support functions in terms of volume, and consider the extension of the Minkowski problem to hedgehogs. We examine hyperbolic \(\mathrm {C}^2\) -hedgehogs in \(\mathbb {R}^{3}\) , proving special cases of a conjecture of A.D. Alexandrov, and giving counterexamples, including discrete versions of hyperbolic hedgehogs and notions of hyperbolicity (weak and strong) for hedgehog polytopes. A strongly hyperbolic polytope arises from discretizing a counterexample to Alexandrov’s conjecture. We give a geometric proof of the Sturm-Hurwitz theorem in the framework of planar multihedgehogs and derive related inequalities. Finally, we solve the Christoffel-Minkowski Problem for general planar hedgehogs, and extend the classical Minkowski inequality (and hence the isoperimetric inequality) to this setting.