Volume, Polar Volume and Euler Characteristic for Convex Functions

Abstract

Functional analogs of the Euler characteristic and volume together with a new analog of the polar volume are characterized as non-negative, continuous, $\mathrm{SL}(n)$ and translation invariant valuations on the space of finite, convex and coercive functions on ${\mathbb{R}}^n$.

References in corpus (7)

• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• Valuations on Convex Functions
• Minkowski valuations on convex functions
• Laplace transforms and valuations
• Affine invariant valuations on polytopes
• Radial continuous rotation invariant valuations on star bodies
• Real-valued valuations on Sobolev spaces

Cited by in corpus (14)

• A homogeneous decomposition theorem for valuations on convex functions
• The support of dually epi-translation invariant valuations on convex functions
• The Hadwiger theorem on convex functions, IV: The Klain approach
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures
• The Hadwiger theorem on convex functions, I
• $\mathrm{SL}(n)$ invariant valuations on super-coercive convex functions
• Valuations on Log-Concave Functions
• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations
• Equivariant Endomorphisms of Convex Functions
• Valuations on Convex Bodies and Functions
• Metrics and Isometries for Convex Functions
• Geometric valuation theory
• A Riesz representation theorem for log-concave functions
• The Legendre transform, the Laplace transform and valuations