# Weakly noncollapsed RCD spaces with upper curvature bounds

## TL;DR

This paper proves that in certain curvature-bounded metric measure spaces, the density function must be constant, linking curvature bounds with measure regularity.

## Contribution

It establishes that $CD(K,n)$ spaces with an upper curvature bound in the Alexandrov sense have constant density functions.

## Key findings

- Density function is constant under the given conditions.
- Curvature bounds impose measure regularity.
- Links between Alexandrov curvature and measure properties.

## Abstract

We show that if a $CD(K,n)$ space $(X,d,f\mathcal{H}^n)$ with $n\geq 2$ has curvature bounded from above by $\kappa$ in the sense of Alexandrov then $f=const$.