What is the size of a function?

### Non-Technical Analogy

The mathematician Stephen Schanuel once asked the question "What is the length of a potato?" The question sounds strange—potatoes are irregularly shaped 3-dimensional objects, and it seems difficult to quantify the "length" of a potato. Length generally makes sense for lines and paths, but Schanuel wanted a concept of length for other objects. Indeed, Schanuel gave a sensible answer to his question, describing a mathematical way of understanding the the "length" of any object.

I answered an analogous question in my Ph.D. thesis: "What is the temperature of a potato?" An object, such as a potato, is usually not the same temperature at every point in its interior, making it difficult to say what is the temperature of a potato. Yet, if we know the temperature at all points inside the potato, can we quantify its average temperature? To use a bit of mathematical terminology, we can think of the temperature of a potato as a function defined at all points in the potato. The question I answered in my thesis is then "What is the size of a function?" I showed that there are various ways of answering this question, and I proved a theorem classifying notions of size for functions.

### A Bit More Detail

A valuation is a way of assigning a notion of size to sets. Hugo Hadwiger classified all valuations (under reasonable assumptions) on compact convex sets: Hadwiger's Theorem says that all such valuations comprise a vector space spanned by the intrinsic volumes. For convex sets in Rn, there are n + 1 intrinsic volumes, which generalize both Lebesgue measure and Euler characteristic. Euler characteristic can be considered a topological notion of size; the other intrinsic volumes are geometric notions of size.

We can generalize the concept of valuation from sets to functions defined on sets. As the intrinsic volumes are valuations on sets, the Hadwiger integrals are valuations on real-valued functions. We can think of the Hadwiger integrals as topological and geometric integrals — or alternately, as topological and geometric notions of size for functions. I have proved a Hadwiger theorem classifying valuations on functions in terms of the Hadwiger integrals. The full theorem requires careful consideration of continuity and topology, and the details are available in my paper. I am now exploring valuations in other contexts, such as simplicial maps, as well as their real-world applications.