Cappello, Anthony R.
(2024)
On the differentiability properties of convex functions and convex bodies.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
This is the latest version of this item.
Abstract
There are three main results in this thesis. We first present a new proof of Theorem 40 that says a convex body K has boundary of class C^{1,1}
if and only if there is R > 0 such that K is the union of closed balls with radius R contained in K. The first main result, Theorem 51 extends the above result to a similar characterization of C^{1,α} convex bodies. Using this characterization, we find new proofs of the Kirchheim-Kristensen theorem (Theorem 70) about the differentiability of the convex envelope and the Krantz-Parks theorem (Theorem 77) about the regularity of the Minkowski sum of convex bodies. Namely we show that if a convex function f : R^n → R satisfies f ∈ C^{1,α}_loc(R^n) and f(x) → ∞ as |x| → ∞, then the convex envelope of f, denoted conv(f), satisfies conv(f) ∈ C^{1,α}_loc(R^n). We also prove that the Minkowski sum of a convex body and a convex body of class C^{1,α} is a convex body of class C^{1,α}. The tools from the characterization of C^{1,1} convex bodies are used to prove the second main result, which is a new geometrically inspired proof of the Alexandrov theorem, Theorem 84, about the second order differentiability of convex functions. Moreover, we give a new proof of a result by Azagra-Hajlasz (Theorem 90) concerning the Lusin Approximation by C^{1,1} convex functions. In the third main result, Theorem 108, we prove the set of normal directions to the k-dimensional faces on the boundary of an n-dimensional convex body is countable (n−k −1)-rectifiable. Finally we conclude by presenting characterizations of C^{1,1} and C^{1,α} functions.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
27 August 2024 |
| Date Type: |
Publication |
| Defense Date: |
30 May 2024 |
| Approval Date: |
27 August 2024 |
| Submission Date: |
5 August 2024 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
116 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
convex body, convex function, Lipschitz gradient, H¨older gradient, convex envelope, Minkowski sum, Alexandrov’s theorem, support function. |
| Date Deposited: |
27 Aug 2024 13:09 |
| Last Modified: |
27 Aug 2024 13:09 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/46873 |
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On the differentiability properties of convex functions and convex bodies. (deposited 27 Aug 2024 13:09)
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