The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality.

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if K is a convex body in R n with 0 ∈ int(K) and if µ is a measure on R n with a locally integrable non-negative density g on R n , then

We prove that if f : R n → [0, ∞) is an integrable log-concave function with f (0) = 1 and F1, . . . , Fr are subspaces of R n such that sIn = r i=1 ciPi where In is the identity operator and Pi is the orthogonal projection onto Fi then n n

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant α n \alpha _n depending (or not) on the dimension n n so that S ( K ) ⩽ α n | K | 1 n max ξ ∈ S n − 1 S ( K ∩ ξ ⊥ ) , \begin{equation*} S(K)\leqslant \alpha _n|K|^{\frac {1}{n}}\max _{\xi \in S^{n-1}}S(K\cap \xi ^{\perp }), \end{equation*} where S S denotes surface area and | ⋅ | |\cdot | denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann–Petty problem with the slicing problem has an affirmative answer.

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