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"The determination of Gauss sums" by Berndt and Evans (Bull. Amer. Math. Soc., Vol. 5, Number 2 (1981), 107-129.) contains an exposition of the original proof due to Gauss. It also includes a short hi …

Random dynamical systems by Ludwig Arnold contains a thorough discussion of various multiplicative ergodic theorems (including the Furstenberg-Kesten result), but not the central limit theorems. As fa …

Fillmore showed that there are sets of constant width in $\mathbb R^d$ with analytic boundaries which have a trivial symmetry group (so these are very different from spheres; see "Symmetries of surfac …

There is a bunch of such statements which can be obtained by Stein's method. You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically …

Grünbaum and Shephard suggest that the winding numbers (for closed polygons) have been discussed in the literature at least since 1769. See A.L.F. Meister, Generalia de genesi figurarum planarum …

A model equation for an inextensible, flexible, heavy surface in a gravitational field was deduced by Poisson Lagrange and later the problem was also studied by Poisson (see the references in the link …

"Topological Vector Spaces" by Helmut Schaefer contains a thorough treatment of the classical Krein-Rutman theorem for compact positive operators in an ordered Banach space along with several general …

This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functi …

Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13). It is based on the classification of real algebra an …

A careful exposition can be found in "Analytic Semigroups and Semilinear Initial Boundary Value Problems" by Kazuaki Taira. He considers the mixed boundary value problem $$\begin{cases} A u=f & \mb …

There exist several known global implicit function theorems. Those results tend to be tailored for specific applications. It seems to be rather difficult to state a universally useful one-size-fits-al …

A proof of the enumeration theorem for the Archimedean solids (which basically dates back to Kepler) can be found in the beautiful book "Polyhedra" by P.R. Cromwell (Cambridge University Press 1997, p …

From the introduction to History of the Central Limit Theorem: From Laplace to Donsker by Hans Fischer: The term “central limit theorem” most likely traces back to Georg Pólya. As he recapitulat …

There is a book devoted entirely to KKM theory and its applications. It contains an outline of the classical KKM theorem as well as a number of generalizations. Another standard reference is "Fixed p …

The rolling motion of a convex symmetric body on a horizontal plane is a classical problem. In the symmetric case, Chaplygin was the first who showed that the full equations of motion can be reduced t …