How to get posts published on the latest date?

You can do it like this:

if ( have_posts() ) :
    /* Queue the first post, that way we know
     * which date we're dealing with.
     *
     * We reset this later so we can run the loop
     * properly with a call to rewind_posts().
     */
    the_post();

    // receive the latest post timestamp
    $latest_post_timestamp = strtotime( get_post()->post_date );

    /* Since we called the_post() above, we need to
     * rewind the loop back to the beginning that way
     * we can run the loop properly, in full.
     */
    rewind_posts();

    // create our custom query and set our latest date arguments
    $the_query = new WP_Query( array(
        'year' => date( 'Y', $latest_post_timestamp ),
        'monthnum' => date( 'm', $latest_post_timestamp ),
        'day' => date( 'd', $latest_post_timestamp ),
    ) );

    // render our latest date posts
    echo '<ul>';
    while ( $the_query->have_posts() ) :
        $the_query->the_post();
        echo '<li>' . get_the_title() . '</li>';
    endwhile;
    echo '</ul>';

    // Restore original query and post data
    wp_reset_query();
    wp_reset_postdata();
endif;