Dembo–Cover–Thomas surface-area inequality

Determine whether every pair of convex bodies A,B in R^n satisfies the Dembo–Cover–Thomas inequality relating the volume-to-surface-area ratio of their Minkowski sum to the corresponding ratios of A and B, namely whether |A+B|/|∂(A+B)|_{n−1} ≥ |A|/|∂A|_{n−1} + |B|/|∂B|_{n−1}.

Background

The paper identifies the Dembo–Cover–Thomas inequality as a principal motivation. The conjecture concerns arbitrary convex bodies in Rn and compares the volume-to-surface-area ratio of a Minkowski sum with the sum of the individual volume-to-surface-area ratios.

Within the class of zonoids, the paper explains that the conjecture is equivalent to a strong projection inequality. The present paper studies related Firey L_p-sum and determinant-power extensions, while the general convex-body conjecture itself remains the motivating unresolved statement.

References

Among their main motivations is a conjecture of Dembo, Cover, and Thomas . They asked whether, for convex bodies A,B\subset\mathbb{R}n, \begin{equation}\label{conj:DCT} \frac{|A+B|}{|\partial(A+B)|{n-1} \geq \frac{|A|}{|\partial A|{n-1} + \frac{|B|}{|\partial B|{n-1}. \end{equation} Here, |\cdot| denotes n-dimensional volume and |\partial K|{n-1} denotes the surface area of K.

conj:DCT:

∣A+B∣∣∂(A+B)∣n−1≥∣A∣∣∂A∣n−1+∣B∣∣∂B∣n−1.\frac{|A+B|}{|\partial(A+B)|_{n-1}} \geq \frac{|A|}{|\partial A|_{n-1}} + \frac{|B|}{|\partial B|_{n-1}}.

— Volume and Projection Inequalities II: Determinants and $L_p$-Sums  (2608.24081 - Fradelizi et al., 25 Aug 2026) in Section 1, Introduction