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Upper semicontinuous valuations on convex functions of one variable

Fernanda M. Baêta

Abstract

A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.

Upper semicontinuous valuations on convex functions of one variable

Abstract

A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on with compact domain.

Paper Structure

This paper contains 3 sections, 23 theorems, 168 equations, 3 figures.

Table of Contents

  1. Introduction and statement of the main result
  2. Tools
  3. Proof of Theorem \ref{['maintheorem']}

Key Result

Theorem 1

A functional $Z:\mathcal{K}^2\rightarrow \mathbb{R}$ is an upper semicontinuous and rigid motion invariant valuation if and only if there exist constants $c_0,c_1,c_2\in\mathbb{R}$ and a function $\zeta\in \mathop{\mathrm{Conc([0,\infty))}}\nolimits$ such that for every $K\in\mathcal{K}^2$.

Figures (3)

  • Figure 1: Quadratic function $q^a$ (gray) and the corresponding approximation $v_n$ (black).
  • Figure 2: Case $u"(x_0)>0$.
  • Figure 3: Case $u"(x_0)=0$.

Theorems & Definitions (39)

  • Theorem 1: 71
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7: 76, Theorem 4.5 (b)
  • ...and 29 more