In this paper, we introduce the concept of complex $L_p$ mixed projection bodies by giving its support function. Then, we establish the complex $L_p$ mixed Petty projection inequalities. Finally, the monotonicity for complex $L_p$ mixed projection bodies is obtained.

All SL(n) equivariant symmetric matrix valued valuations on convex polytopes in R-n are completely classified without any continuity assumptions. The unique ones turn out to be the moment matrices corresponding to the classical Legendre ellipsoid and the isotropic position.

All $$\mathrm{SL}(n)$$ contravariant vector valuations on polytopes in $${\mathbb {R}}^n$$ are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique class of such valuations for $$n\ge 3$$ . In dimension two, the classification corresponds to the known case of $$SL (2)$$ covariant valuations.

All SL(n) contravariant symmetric matrix valued valuations on convex polytopes in R-n are completely classified without any continuity assumptions. The general Lutwak-Yang-Zhang matrix is shown to be essentially the unique such valuation.

It is proved that the classical Laplace transform is a continuous valuation which is positively GL(n) covariant and logarithmic translation covariant. Conversely, these properties turn out to be sufficient to characterize this transform.

All SL(n) covariant vector valuations on convex polytopes in R-n are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if n >= 3, while two new functionals show up in dimension two.

It is proved that the classical Laplace transform is a continuous valuation which is positively GL(n) covariant and logarithmic translation covariant. Conversely, these properties turn out to be sufficient to characterize this transform.

Continuous, SL(n) and translation invariant real-valued valuations on Sobolev spaces are classified. The centro-affine Hadwiger's theorem is applied. In the homogeneous case, these valuations turn out to be L (p) -norms raised to p-th power (up to suitable multipication scales).

Bourgain, Brezis, and Mironescu showed that (with suitable scaling) the fractional Sobolev s-seminorm of a function \({f \in W^{1,p}(\mathbb{R}^n)}\) converges to the Sobolev seminorm of f as \({s\rightarrow1^-}\) . Ludwig introduced the anisotropic fractional Sobolev s-seminorms of f defined by a norm on \({\mathbb{R}^n}\) with unit ball K and showed that they converge to the anisotropic Sobolev seminorm of f defined by the norm whose unit ball is the polar L p moment body of K, as \({s \rightarrow 1^-}\) . The asymmetric anisotropic s-seminorms are shown to converge to the anisotropic Sobolev seminorm of f defined by the Minkowski functional of the polar asymmetric L p moment body of K.

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k 0,\ 1\le k

The problem of finding the maximal hyperplane section of Bpn, where p>2, has been open for a long time. It is known that the answer depends on both p and n. In this paper, using the well-known equivalence between hyperplane sections and the isotropic constant of a body, we give an upper bound estimate for the volume of hyperplane sections of normalized ℓpn-balls that does not depend on n and p. In addition, on the basis of results of Meyer, Pajor and Schmuckenschläger, we show further the corresponding extremal body and hyperplane section when this volume attains its minimum.