Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger-Kazdan inequality and Santaló's formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

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 $\mathcal{H}_{h}$ is called the hedgehog with support function $h$. More generally, the object of hedgehog theory is to study the Brunn--Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space $\mathbb{R} ^{n+1}$, defined as (possibly singular and self-intersecting) hypersurfaces of $\mathbb{R}^{n+1}$. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their length measures and solve the extension of the Christoffel--Minkowski problem to plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in $\mathbb{R}^{2}$ and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also present an isoperimetric inequality in the Minkowski spacetime $\mathbb{L}^{n+1}$ as a consequence of this Brunn-Minkowski type theorem.