Final Oral Public Examination

A core concept in convex geometry is the mixed volume of convex bodies, a geometric quantity with rich connections to various other fields such as intersection theory and enumerative combinatorics. Numerous inequalities for mixed volumes are known; they can be used to prove inequalities arising in algebraic geometry and combinatorics, and often vice versa. This thesis consists of three works around mixed volumes of the author during his PhD.

 

In the first part, we study the monotonicity problem for mixed volumes: when does the mixed volume increase strictly as one convex body becomes larger? This is a problem posed by Minkowski in his foundational work on the theory of mixed volumes and remained largely open even in dimension 3. Confirming a conjecture of Schneider, we prove an upper bound on the support of the mixed area measure, and we show that this bound is sharp in R3 as well as in several other settings. As a consequence, we also obtain a natural extension of a classical theorem of Hartman–Nirenberg in differential geometry about ruled surfaces.

 

In the second part, we investigate the local log-Brunn–Minkowski (LLBM) inequality. We develop a monotonicity principle for the LLBM deficit under Minkowski addition of line segments. On the one hand, this yields a new proof of LLBM for zonoids. On the other hand, assuming the validity of LLBM, it implies that for smooth origin-symmetric bodies with C2 support functions, the only equality cases are the expected trivial ones.

 

In the third part, we investigate projection areas of convex bodies in R4, motivated by the Heine–Shephard problem of giving numerical characterizations of mixed volumes. We prove that a sextuple of numbers arises as the areas of the six coordinate 2-dimensional projections of a convex body if and only if it satisfies a Plücker-type relation over triangular hyperfields. We further show that this condition is equivalent to the Lorentzian property of an associated matrix determined by the sextuple, thereby resolving the realizability of the corresponding Lorentzian polynomial as a volume polynomial.